what is problem solving function

Module 5: Function Basics

Evaluating and solving functions, learning outcomes.

• Evaluate and solve functions in algebraic form.
• Evaluate functions given tabular or graphical data.

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f\left(x\right)=5 - 3{x}^{2}[/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

How To: EVALUATE A FUNCTION Given ITS FORMula.

• Replace the input variable in the formula with the value provided.
• Calculate the result.

Example: Evaluating Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], evaluate [latex]h\left(4\right)[/latex].

To evaluate [latex]h\left(4\right)[/latex], we substitute the value 4 for the input variable [latex]p[/latex] in the given function.

[latex]\begin{align}h\left(p\right)&={p}^{2}+2p \\ h\left(4\right)&={\left(4\right)}^{2}+2\left(4\right) \\ &=16+8 \\ &=24 \end{align}[/latex]

Therefore, for an input of 4, we have an output of 24 or [latex]h(4)=24[/latex].

In the following video we offer more examples of evaluating a function for specific x values.

Example: Evaluating Functions at Specific Values

For the function, [latex]f\left(x\right)={x}^{2}+3x - 4[/latex], evaluate each of the following.

• [latex]f\left(2\right)[/latex]
• [latex]f(a)[/latex]
• [latex]f(a+h)[/latex]
• [latex]\dfrac{f\left(a+h\right)-f\left(a\right)}{h}[/latex]

Replace the [latex]x[/latex] in the function with each specified value.

• Because the input value is a number, 2, we can use algebra to simplify. [latex]\begin{align}f\left(2\right)&={2}^{2}+3\left(2\right)-4 \\ &=4+6 - 4 \\ &=6\hfill \end{align}[/latex]
• In this case, the input value is a letter so we cannot simplify the answer any further. [latex]f\left(a\right)={a}^{2}+3a - 4[/latex]
• With an input value of [latex]a+h[/latex], we must use the distributive property. [latex]\begin{align}f\left(a+h\right)&={\left(a+h\right)}^{2}+3\left(a+h\right)-4 \\[2mm] &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \end{align}[/latex]

and we know that

Now we combine the results and simplify.

[latex]\begin{align}\dfrac{f\left(a+h\right)-f\left(a\right)}{h}&=\dfrac{\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\right)-\left({a}^{2}+3a - 4\right)}{h} \\[2mm] &=\dfrac{2ah+{h}^{2}+3h}{h}\\[2mm] &=\frac{h\left(2a+h+3\right)}{h}&&\text{Factor out }h. \\[2mm] &=2a+h+3&&\text{Simplify}.\end{align}[/latex]

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], evaluate [latex]g\left(5\right)[/latex].

[latex]g\left(5\right)=\sqrt{5 - 4}=1[/latex]

In the next video, we provide another example of how to solve for a function value.

Example: Solving Functions

Given the function [latex]h\left(p\right)={p}^{2}+2p[/latex], solve for [latex]h\left(p\right)=3[/latex].

[latex]\begin{align}&h\left(p\right)=3\\ &{p}^{2}+2p=3 &&\text{Substitute the original function }h\left(p\right)={p}^{2}+2p. \\ &{p}^{2}+2p - 3=0 &&\text{Subtract 3 from each side}. \\ &\left(p+3\text{)(}p - 1\right)=0 &&\text{Factor}. \end{align}[/latex]

If [latex]\left(p+3\right)\left(p - 1\right)=0[/latex], either [latex]\left(p+3\right)=0[/latex] or [latex]\left(p - 1\right)=0[/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[/latex] in each case.

[latex]\begin{align}&p+3=0, &&p=-3 \\ &p - 1=0, &&p=1\hfill \end{align}[/latex]

This gives us two solutions. The output [latex]h\left(p\right)=3[/latex] when the input is either [latex]p=1[/latex] or [latex]p=-3[/latex].

We can also verify by graphing as in Figure 5. The graph verifies that [latex]h\left(1\right)=h\left(-3\right)=3[/latex] and [latex]h\left(4\right)=24[/latex].

Given the function [latex]g\left(m\right)=\sqrt{m - 4}[/latex], solve [latex]g\left(m\right)=2[/latex].

[latex]m=8[/latex]

Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[/latex] expresses a functional relationship between [latex]n[/latex] and [latex]p[/latex]. We can rewrite it to decide if [latex]p[/latex] is a function of [latex]n[/latex].

How To: Given a function in equation form, write its algebraic formula.

• Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.
• Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example: Finding an Equation of a Function

Express the relationship [latex]2n+6p=12[/latex] as a function [latex]p=f\left(n\right)[/latex], if possible.

To express the relationship in this form, we need to be able to write the relationship where [latex]p[/latex] is a function of [latex]n[/latex], which means writing it as [latex]p=[/latex] expression involving [latex]n[/latex].

[latex]\begin{align}&2n+6p=12\\[1mm] &6p=12 - 2n &&\text{Subtract }2n\text{ from both sides}. \\[1mm] &p=\frac{12 - 2n}{6} &&\text{Divide both sides by 6 and simplify}. \\[1mm] &p=\frac{12}{6}-\frac{2n}{6} \\[1mm] &p=2-\frac{1}{3}n \end{align}[/latex]

Therefore, [latex]p[/latex] as a function of [latex]n[/latex] is written as

[latex]p=f\left(n\right)=2-\frac{1}{3}n[/latex]

Analysis of the Solution

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Watch this video to see another example of how to express an equation as a function.

Example: Expressing the Equation of a Circle as a Function

Does the equation [latex]{x}^{2}+{y}^{2}=1[/latex] represent a function with [latex]x[/latex] as input and [latex]y[/latex] as output? If so, express the relationship as a function [latex]y=f\left(x\right)[/latex].

First we subtract [latex]{x}^{2}[/latex] from both sides.

[latex]{y}^{2}=1-{x}^{2}[/latex]

We now try to solve for [latex]y[/latex] in this equation.

[latex]\begin{align}y&=\pm \sqrt{1-{x}^{2}} \\[1mm] &=\sqrt{1-{x}^{2}}\hspace{3mm}\text{and}\hspace{3mm}-\sqrt{1-{x}^{2}} \end{align}[/latex]

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f\left(x\right)[/latex].

If [latex]x - 8{y}^{3}=0[/latex], express [latex]y[/latex] as a function of [latex]x[/latex].

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation [latex]x=y+{2}^{y}[/latex], if we want to express [latex]y[/latex] as a function of [latex]x[/latex], there is no simple algebraic formula involving only [latex]x[/latex] that equals [latex]y[/latex]. However, each [latex]x[/latex] does determine a unique value for [latex]y[/latex], and there are mathematical procedures by which [latex]y[/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for [latex]y[/latex] as a function of [latex]x[/latex], even though the formula cannot be written explicitly.

Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.

The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.

At times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[/latex].

The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function [latex]P[/latex] at the input value of “goldfish.” We would write [latex]P\left(\text{goldfish}\right)=2160[/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.

How To: Given a function represented by a table, identify specific output and input values.

• Find the given input in the row (or column) of input values.
• Identify the corresponding output value paired with that input value.
• Find the given output values in the row (or column) of output values, noting every time that output value appears.
• Identify the input value(s) corresponding to the given output value.

Example: Evaluating and Solving a Tabular Function

Using the table below,

• Evaluate [latex]g\left(3\right)[/latex].
• Solve [latex]g\left(n\right)=6[/latex].
• Evaluating [latex]g\left(3\right)[/latex] means determining the output value of the function [latex]g[/latex] for the input value of [latex]n=3[/latex]. The table output value corresponding to [latex]n=3[/latex] is 7, so [latex]g\left(3\right)=7[/latex].
• Solving [latex]g\left(n\right)=6[/latex] means identifying the input values, [latex]n[/latex], that produce an output value of 6. The table below shows two solutions: [latex]n=2[/latex] and [latex]n=4[/latex].

When we input 2 into the function [latex]g[/latex], our output is 6. When we input 4 into the function [latex]g[/latex], our output is also 6.

Using the table from the previous example, evaluate [latex]g\left(1\right)[/latex] .

Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

Example: Reading Function Values from a Graph

Given the graph below,

• Evaluate [latex]f\left(2\right)[/latex].
• Solve [latex]f\left(x\right)=4[/latex].

Using the graph, solve [latex]f\left(x\right)=1[/latex].

[latex]x=0[/latex] or [latex]x=2[/latex]

• Graph the function [latex]f(x) = -\frac{1}{2}x^2+x+4[/latex] using function notation.
• Evaluate the function at [latex]x=1[/latex]
• Make a table of values that references the function. Include at least the interval [latex][-5,5][/latex] for [latex]x[/latex]-values.
• Solve the function for [latex]f(0)[/latex]

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17.2.1: Evaluating Functions

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Learning Objectives

• Given a function, described by an equation, find function values (outputs) for specified inputs.

Introduction

Throughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, \(\ y=4 x+1\) is an equation that represents a function. When you input values for \(\ x\), you can determine a single output for \(\ y\). In this case, if you substitute \(\ x=10\) into the equation you will find that \(\ y\) must be 41; there is no other value of \(\ y\) that would make the equation true.

Rather than using the variable \(\ y\), the equations of functions can be written using function notation . Function notation is very useful when you are working with more than one function at a time, and substituting more than one variable in for \(\ x\).

Function Notation

Some people think of functions as “mathematical machines.” Imagine you have a machine that changes a number according to a specific rule, such as “multiply by 3 and add 2” or “divide by 5, add 25, and multiply by -1.” If you put a number into the machine, a new number will pop out the other end, having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.

You can also call the machine “\(\ f\)” for function. If you put \(\ x\) into the machine, \(\ f(x)\) comes out. Mathematically speaking, \(\ x\) is the input, or the “independent variable,” and \(\ f(x)\) is the output, or the “dependent variable,” since it depends on the value of \(\ x\).

\(\ f(x)=4 x+1\) is written in function notation and is read “\(\ f\) of \(\ x\) equals \(\ 4x\) plus 1.” It represents the following situation: A function named \(\ f\) acts upon an input, \(\ x\), and produces \(\ f(x)\) which is equal to \(\ 4 x+1\). This is the same as the equation \(\ y=4 x+1\).

Function notation gives you more flexibility because you don’t have to use \(\ y\) for every equation. Instead, you can use \(\ f(x)\) or \(\ g(x)\) or \(\ c(x)\). This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.

You could write the formula for perimeter, \(\ P=4 s\), as the function \(\ p(x)=4 x\), and the formula for area, \(\ A=x^{2}\), as \(\ a(x)=x^{2}\). This would make it easy to graph both functions on the same graph without confusion about the variables.

Which two equations represent the same function?

• \(\ y=2 x-7 \text { and } f(x)=7-2 x\)
• \(\ 3 x=y-2 \text { and } f(x)=3 x-2\)
• \(\ f(x)=3 x^{2}+5 \text { and } y=3 x^{2}+5\)
• None of the above
• Incorrect. These equations look similar but are not the same. The first has a slope of 2 and a y-intercept of -7. The second function has a slope of -2 and a y-intercept of 7. It slopes in the opposite direction. They do not produce the same graph, so they are not the same function. The correct answer is \(\ f(x)=3 x^{2}+5\) and \(\ y=3 x^{2}+5\).
• Incorrect. These equations represent two different functions. If you rewrite the first equation in terms of \(\ y\), you’ll find the equation of the function is \(\ y=3 x+2\). The correct answer is \(\ f(x)=3 x^{2}+5\) and \(\ y=3 x^{2}+5\).
• Correct. The expressions after \(\ f(x)=\) and \(\ y=\) are the same, so these are two different ways to write the same function: \(\ f(x)=3 x^{2}+5\) and \(\ y=3 x^{2}+5\).
• Incorrect. Look at the expressions that follow \(\ f(x)=\) and \(\ y=\). If the expressions are the same, then the equations represent the same exact function. The correct answer is \(\ f(x)=3 x^{2}+5\) and \(\ y=3 x^{2}+5\).

Evaluating Functions

Equations written using function notation can also be evaluated. With function notation, you might see a problem like this.

Given \(\ f(x)=4 x+1\), find \(\ f(2)\).

You read this problem like this: “given \(\ f\) of \(\ x\) equals \(\ 4x\) plus one, find \(\ f\) of 2.” While the notation and wording is different, the process of evaluating a function is the same as evaluating an equation: in both cases, you substitute 2 for \(\ x\), multiply it by 4 and add 1, simplifying to get 9. In both a function and an equation, an input of 2 results in an output of 9.

\(\ \begin{array}{l} f(x)=4 x+1 \\ f(2)=4(2)+1=8+1=9 \end{array}\)

You can simply apply what you already know about evaluating expressions to evaluate a function. It’s important to note that the parentheses that are part of function notation do not mean multiply. The notation \(\ f(x)\) does not mean \(\ f\) multiplied by \(\ x\). Instead the notation means “\(\ f\) of \(\ x\)” or “the function of \(\ x\)” To evaluate the function, take the value given for \(\ x\), and substitute that value in for \(\ x\) in the expression. Let’s look at a couple of examples.

Given \(\ f(x)=3 x-4\), find \(\ f(5)\).

Given \(\ f(x)=3 x-4, f(5)=11\).

Functions can be evaluated for negative values of \(\ x\), too. Keep in mind the rules for integer operations.

Given \(\ p(x)=2 x^{2}+5\), find \(\ p(-3)\).

Given \(\ p(x)=2 x^{2}+5, p(-3)=23\).

You may also be asked to evaluate a function for more than one value as shown in the example that follows.

Given \(\ f(x)=3 x^{2}+2 x+1\), find \(\ f(0)\), \(\ f(2)\), and \(\ f(-1)\).

Given \(\ f(x)=3 x^{2}+2 x+1\), \(\ f(0)=1\), \(\ f(2)=17\), and \(\ f(-1)=2\).

Given \(\ h(x)=4 x+7\), find \(\ h(-10)\).

• \(\ -40 h+7\)
• \(\ 4 x+17\)
• Incorrect. \(\ h(-10)\) means “\(\ h\) of negative ten” not “\(\ h\) times negative ten.” To evaluate the function, substitute \(\ -10\) for \(\ x\). The correct answer is \(\ -33\).
• Correct. \(\ h(-10)=4(-10)+7=-40+7=-33\).
• Incorrect. To find \(\ h(-10)\), substitute \(\ -10\) in for \(\ x\) in the right side of the equation and simplify. The correct answer is \(\ -33\).
• Incorrect. Evaluate the function for \(\ h(-10)\), not \(\ h(10)\). The correct answer is \(\ -33\).

Evaluating Functions with Variable Inputs

So far, you have evaluated functions for inputs that have been constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.

Given \(\ f(x)=3 x^{2}+2 x+1\), find \(\ f(b)\).

Given \(\ f(x)=3 x^{2}+2 x+1\), \(\ f(b)=3 b^{2}+2 b+1\).

In the following example, you evaluate a function for an expression. So here you will substitute the entire expression in for \(\ x\) and simplify.

Given \(\ f(x)=4 x+1\), find \(\ f(h+1)\).

Given \(\ f(x)=4 x+1, f(h+1)=4 h+5\).

Function notation takes the form such as \(\ f(x)=18 x-10\) and is read “\(\ f\) of \(\ x\) equals 18 times \(\ x\) minus 10.” Function notation can use letters other than \(\ f\), such as \(\ c(x)\), \(\ g(x)\), or \(\ h(x)\). As you go further in your study of functions, this notation will provide you more flexibility, allowing you to examine and compare different functions more easily. Just as an algebraic equation written in \(\ x\) and \(\ y\) can be evaluated for different values of the input \(\ x\), an equation written in function notation can also be evaluated for different values of \(\ x\). To evaluate a function, substitute in values for \(\ x\) and simplify to find the related output.

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

What Is Problem Solving? How Software Engineers Approach Complex Challenges

From debugging an existing system to designing an entirely new software application, a day in the life of a software engineer is filled with various challenges and complexities. The one skill that glues these disparate tasks together and makes them manageable? Problem solving . 

Throughout this blog post, we’ll explore why problem-solving skills are so critical for software engineers, delve into the techniques they use to address complex challenges, and discuss how hiring managers can identify these skills during the hiring process. 

What Is Problem Solving?

But what exactly is problem solving in the context of software engineering? How does it work, and why is it so important?

Problem solving, in the simplest terms, is the process of identifying a problem, analyzing it, and finding the most effective solution to overcome it. For software engineers, this process is deeply embedded in their daily workflow. It could be something as simple as figuring out why a piece of code isn’t working as expected, or something as complex as designing the architecture for a new software system. 

In a world where technology is evolving at a blistering pace, the complexity and volume of problems that software engineers face are also growing. As such, the ability to tackle these issues head-on and find innovative solutions is not only a handy skill — it’s a necessity. 

The Importance of Problem-Solving Skills for Software Engineers

Problem-solving isn’t just another ability that software engineers pull out of their toolkits when they encounter a bug or a system failure. It’s a constant, ongoing process that’s intrinsic to every aspect of their work. Let’s break down why this skill is so critical.

Driving Development Forward

Without problem solving, software development would hit a standstill. Every new feature, every optimization, and every bug fix is a problem that needs solving. Whether it’s a performance issue that needs diagnosing or a user interface that needs improving, the capacity to tackle and solve these problems is what keeps the wheels of development turning.

It’s estimated that 60% of software development lifecycle costs are related to maintenance tasks, including debugging and problem solving. This highlights how pivotal this skill is to the everyday functioning and advancement of software systems.

Innovation and Optimization

The importance of problem solving isn’t confined to reactive scenarios; it also plays a major role in proactive, innovative initiatives . Software engineers often need to think outside the box to come up with creative solutions, whether it’s optimizing an algorithm to run faster or designing a new feature to meet customer needs. These are all forms of problem solving.

Consider the development of the modern smartphone. It wasn’t born out of a pre-existing issue but was a solution to a problem people didn’t realize they had — a device that combined communication, entertainment, and productivity into one handheld tool.

Increasing Efficiency and Productivity

Good problem-solving skills can save a lot of time and resources. Effective problem-solvers are adept at dissecting an issue to understand its root cause, thus reducing the time spent on trial and error. This efficiency means projects move faster, releases happen sooner, and businesses stay ahead of their competition.

Improving Software Quality

Problem solving also plays a significant role in enhancing the quality of the end product. By tackling the root causes of bugs and system failures, software engineers can deliver reliable, high-performing software. This is critical because, according to the Consortium for Information and Software Quality, poor quality software in the U.S. in 2022 cost at least $2.41 trillion in operational issues, wasted developer time, and other related problems.

Problem-Solving Techniques in Software Engineering

So how do software engineers go about tackling these complex challenges? Let’s explore some of the key problem-solving techniques, theories, and processes they commonly use.

Decomposition

Breaking down a problem into smaller, manageable parts is one of the first steps in the problem-solving process. It’s like dealing with a complicated puzzle. You don’t try to solve it all at once. Instead, you separate the pieces, group them based on similarities, and then start working on the smaller sets. This method allows software engineers to handle complex issues without being overwhelmed and makes it easier to identify where things might be going wrong.

Abstraction

In the realm of software engineering, abstraction means focusing on the necessary information only and ignoring irrelevant details. It is a way of simplifying complex systems to make them easier to understand and manage. For instance, a software engineer might ignore the details of how a database works to focus on the information it holds and how to retrieve or modify that information.

Algorithmic Thinking

At its core, software engineering is about creating algorithms — step-by-step procedures to solve a problem or accomplish a goal. Algorithmic thinking involves conceiving and expressing these procedures clearly and accurately and viewing every problem through an algorithmic lens. A well-designed algorithm not only solves the problem at hand but also does so efficiently, saving computational resources.

Parallel Thinking

Parallel thinking is a structured process where team members think in the same direction at the same time, allowing for more organized discussion and collaboration. It’s an approach popularized by Edward de Bono with the “ Six Thinking Hats ” technique, where each “hat” represents a different style of thinking.

In the context of software engineering, parallel thinking can be highly effective for problem solving. For instance, when dealing with a complex issue, the team can use the “White Hat” to focus solely on the data and facts about the problem, then the “Black Hat” to consider potential problems with a proposed solution, and so on. This structured approach can lead to more comprehensive analysis and more effective solutions, and it ensures that everyone’s perspectives are considered.

This is the process of identifying and fixing errors in code . Debugging involves carefully reviewing the code, reproducing and analyzing the error, and then making necessary modifications to rectify the problem. It’s a key part of maintaining and improving software quality.

Testing and Validation

Testing is an essential part of problem solving in software engineering. Engineers use a variety of tests to verify that their code works as expected and to uncover any potential issues. These range from unit tests that check individual components of the code to integration tests that ensure the pieces work well together. Validation, on the other hand, ensures that the solution not only works but also fulfills the intended requirements and objectives.

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Evaluating Problem-Solving Skills

We’ve examined the importance of problem-solving in the work of a software engineer and explored various techniques software engineers employ to approach complex challenges. Now, let’s delve into how hiring teams can identify and evaluate problem-solving skills during the hiring process.

Recognizing Problem-Solving Skills in Candidates

How can you tell if a candidate is a good problem solver? Look for these indicators:

• Previous Experience: A history of dealing with complex, challenging projects is often a good sign. Ask the candidate to discuss a difficult problem they faced in a previous role and how they solved it.
• Problem-Solving Questions: During interviews, pose hypothetical scenarios or present real problems your company has faced. Ask candidates to explain how they would tackle these issues. You’re not just looking for a correct solution but the thought process that led them there.
• Technical Tests: Coding challenges and other technical tests can provide insight into a candidate’s problem-solving abilities. Consider leveraging a platform for assessing these skills in a realistic, job-related context.

Assessing Problem-Solving Skills

Once you’ve identified potential problem solvers, here are a few ways you can assess their skills:

• Solution Effectiveness: Did the candidate solve the problem? How efficient and effective is their solution?
• Approach and Process: Go beyond whether or not they solved the problem and examine how they arrived at their solution. Did they break the problem down into manageable parts? Did they consider different perspectives and possibilities?
• Communication: A good problem solver can explain their thought process clearly. Can the candidate effectively communicate how they arrived at their solution and why they chose it?
• Adaptability: Problem-solving often involves a degree of trial and error. How does the candidate handle roadblocks? Do they adapt their approach based on new information or feedback?

Hiring managers play a crucial role in identifying and fostering problem-solving skills within their teams. By focusing on these abilities during the hiring process, companies can build teams that are more capable, innovative, and resilient.

Key Takeaways

As you can see, problem solving plays a pivotal role in software engineering. Far from being an occasional requirement, it is the lifeblood that drives development forward, catalyzes innovation, and delivers of quality software. 

By leveraging problem-solving techniques, software engineers employ a powerful suite of strategies to overcome complex challenges. But mastering these techniques isn’t simple feat. It requires a learning mindset, regular practice, collaboration, reflective thinking, resilience, and a commitment to staying updated with industry trends. 

For hiring managers and team leads, recognizing these skills and fostering a culture that values and nurtures problem solving is key. It’s this emphasis on problem solving that can differentiate an average team from a high-performing one and an ordinary product from an industry-leading one.

At the end of the day, software engineering is fundamentally about solving problems — problems that matter to businesses, to users, and to the wider society. And it’s the proficient problem solvers who stand at the forefront of this dynamic field, turning challenges into opportunities, and ideas into reality.

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Executive Functioning Skills 101: Problem-Solving

Written by:

  Amy Sippl

Filed under: Problem Solving , Behaviors , EF 101 Series , Executive Functioning

Published:  December 24, 2020

Last Reviewed: February 13, 2024

READING TIME:  ~ minutes

When parents reach out to us about working with a child, we almost always ask about problem solving skills.

• Does your child identify problems?
• When you see a problem in the making, can they see it too?
• If they come to you with a problem, how much effort does it take to generate a solution?
• Can they evaluate the pros and cons of different strategies to solve a problem?
• When your child eventually finds a solution to a problem, can they implement it and stay the course to see the situation through?

For many of the parents and teachers we work with, it quickly becomes clear that the struggles and challenging behaviors they encounter with their child ultimately comes back to challenges with problem solving.

What is problem solving?

Problem solving involves our capacity to identify and describe a problem and generate solutions to fix it .

According to Newell & Simon (1972) , “a person is confronted with a problem when he wants something and does not know immediately what series of actions he can perform to get it” (p.72). We engage in problem solving behaviors when we evaluate possible steps forward and then take action to obtain an outcome.

Problem solving involves many other executive functioning behaviors, including attentional control, planning , and task initiation . We need to pay attention to our environment to notice a problem, outline different strategies, and then attempt one of those solutions. Depending on the issue, we might also need to use time management , emotional control , or organization skills. Over time, if we can observe our behavior and understanding of the environment through working memory and self-monitoring behaviors, it can also influence our problem solving skills.

Developing Problem Solving Behaviors

Problem solving skills develop early through play behaviors as infants and toddlers. At these early stages of development, much of a child’s play consists of cause and effect activities or ‘figuring out how things work.’ As children move into early learning years, problem solving includes learning decision making and turn-taking. Children learn to brainstorm solutions to simple problems and learn to notice issues when others point them out.

As children reach adolescence, they begin to independently identify problems in many settings, including home, school, work, and friends. They sort out conflicts and decide what steps to take but may seek adult feedback and support to evaluate the potential advantages and disadvantages. Developing good problem solving skills as adults involves generating unique solutions to complex problems and persisting through multiple solutions until a problem resolves.

Examples of Problem Solving

How we problem solve might look different depending on the situation. Here are several examples of skills we need to know to problem solve effectively:

• Complete puzzles and games to accomplish a goal.
• Use language and body movements to achieve an outcome.
• Identify and define a problem, including where the problem originated and why.
• Break apart a problem into smaller parts.
• Identify problems in different social contexts, including work, school, home, and friends.
• Sort out conflicts and what to do in social situations.
• Seek guidance from others about what to do.
• Persist in developing new strategies when previous attempts fail.

Problem Solving and Challenging Behavior

Not all teens and young adults develop strong problem solving behaviors like those listed above. Barriers related to diagnosis, learning history, and motivation can impact how a child or student handles problems when they arise.

Learners with a history of behavioral and learning challenges may not always use good problem solving skills to manage stressful situations. We all know students who use challenging behaviors like talking back, arguing, property destruction, and aggression when presented with challenging tasks. We also know students who shut down, check out, or struggle to follow directions when encountering new or unknown situations. Without a step-by-step model for problem solving, including identifying a problem and choosing a replacement behavior to solve it, many of our children and students use challenging behaviors instead. Over time, learners written off as ‘defiant,’ ‘lazy,’ or ‘helpless’ may just need better problem solving tools.

How to Evaluate Problem Solving Issues

Some of the negative consequences of when a child or student struggles with problem solving skills might be obvious. Still, it may be more difficult to evaluate where there are underlying areas of concern. Here are some strategies to consider if it’s time to intervene:

• Conduct a behavioral observation. Using the list of behaviors above, observe your child or student when they encounter a problem. Jot down strengths and areas where the process stumbles. Notice what they solve independently and the types of issues that consistently require support.
• Conduct a skills assessment. There are many different tools, checklists, and workbooks (get 20% off our executive functioning workbook with coupon code LSA20 ) available to evaluate and create goals around executive functioning skills like organization . Many of these assessments and evaluation tools can also help develop SMART goals to target down the road. Download our problem solving skills pre-assessment below and complete it with your child.
• Meet with your child’s care team. Chances are if you’re noticing issues with problem solving skills, your child’s caregivers, teachers, coaches, and family members are also observing similar things. Gather the other stakeholders involved to coordinate and learn more about where problems arise.
• Contact a professional or life skills coach. Not every parent or teacher comes equipped to address problem solving skills. Working with a trained professional with experience in helping children develop and enhance problem solving can help teach new skills more efficiently and effectively.

Further Reading

• Bransford, J. D., and Stein, B. S., “The Ideal Problem Solver” (1993). Centers for Teaching and Technology – Book Library . 46.
• Kern, L., George, M. P., & Weist, M. D. (2016). Supporting students with emotional and behavioral problems.  Baltimore, MD: Paul H. Brookes .
• Life Skills Advocate Blog: Executive Function Skills by Age: What to Look For
• Life Skills Advocate Blog: How Teaching Executive Functioning Skills Can Reduce Challenging Behaviors
• Life Skills Advocate Blog: SMART Goal Setting
• Newell, A., & Simon, H. A. (1972).  Human problem solving  (Vol. 104, No. 9). Englewood Cliffs, NJ: Prentice-Hall.

About The Author

Amy Sippl is a Minnesota-based Board Certified Behavior Analyst (BCBA) and freelance content developer specializing in helping individuals with autism and their families reach their best possible outcomes. Amy earned her Master's Degree in Applied Behavior Analysis from St. Cloud State University and also holds undergraduate degrees in Psychology and Family Social Science from University of Minnesota – Twin Cities. Amy has worked with children with autism and related developmental disabilities for over a decade in both in-home and clinical settings. Her content focuses on parents, educators, and professionals in the world of autism—emphasizing simple strategies and tips to maximize success. To see more of her work visit amysippl.com .

Related Posts

Understanding the executive functioning ripple effect, finding focus: how to help your teen with paying attention, how to make vacation planning executive function friendly, 5 hidden benefits of executive function coaching, ai is changing the executive functioning landscape: here’s what you should know, executive functioning skills 101: the basics of task initiation.

Life Skills Advocate is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com. Some of the links in this post may be Amazon.com affiliate links, which means if you make a purchase, Life Skills Advocate will earn a commission. However, we only promote products we actually use or those which have been vetted by the greater community of families and professionals who support individuals with diverse learning needs.

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What Are Cognitive Functions?

Memory, speed, reasoning, language abilities, and more..

Posted June 28, 2020 | Reviewed by Jessica Schrader

In psychological health research, a person’s ability to think, otherwise known as their cognitive functions, is a crucial subject of research. The cognitive functions are a variety of different, but related, skills involving learning and problem-solving. Together, these are some of the most important and impressive abilities that the brain is capable of.

The cognitive functions are collectively known by several different names. Different researchers with different expertise call them general cognitive function, cognitive ability, cognitive capacity, and intelligence. This is the same “intelligence” of the intelligence quotient , better known as IQ. High-quality IQ tests measure the same thing as what health psychologists might call general cognitive function. Because standardized IQ tests are commonly given in school, they have been repurposed by health researchers to measure a person’s pre-morbid cognitive function.

In other words, if you measure cognitive function early in life before major health problems arise that might impact cognitive function, you can make stronger claims about whether these health problems predict cognitive dysfunction. This work is part of a subfield called cognitive epidemiology .

The individual cognitive functions themselves are numerous and diverse . Many make more intuitive sense than general cognitive function, which is vague. The memory domain represents your ability to remember things. The speed domain represents how fast you can think. Others are not as intuitive. Reasoning represents your ability to solve new problems, particularly with nonverbal information, and verbal fluency represents your ability to work with verbal information. Visual-spatial ability represents your proficiency in interacting with physical objects in the world. Humans rely heavily on our sense of sight, but we also are unique in how good we are at manipulating our physical environments and doing things like making tools. Hence, humans have developed distinctly strong visual-spatial abilities.

The individual cognitive functions themselves are not perfectly distinct, however. For instance, memory can be subdivided into episodic memory, which assesses your ability to remember details like items in a list or events from a story, and working memory, which is your ability to simultaneously keep information in mind, like when you try to remember a telephone number. Though all cognitive abilities tend to be correlated, that is, if you are good at one thing you tend to be good at all the others, these memory abilities tend to be more strongly correlated to each other than to other cognitive abilities, e.g., speed.

We are starting to learn how the individual functions are important for health, as well. For instance, take memory again. All of these abilities tend to decline as you age. If you are healthy, the decline is slow. In certain cases, the decline is faster, however. In people with Alzheimer’s disease, for example, the memory domain in particular declines more rapidly than we would expect. You can observe these declines in memory before you can classify a person as having dementia . Ideally, we can find more ways to use tests of cognitive function to reliably identify psychological health problems sooner than we would otherwise. When used properly, tests of the cognitive functions are a powerful healthcare tool with wide applications.

Deary, I. J., & Batty, G. D. (2007). Cognitive epidemiology. Journal of Epidemiology & Community Health, 61(5), 378-384.

Dowling, N. M., Farias, S. T., Reed, B. R., Sonnen, J. A., Strauss, M. E., Schneider, J. A., ... & Mungas, D. (2011). Neuropathological associates of multiple cognitive functions in two community-based cohorts of older adults. Journal of the International Neuropsychological Society: JINS, 17(4), 602.

Rabin, L. A., Paré, N., Saykin, A. J., Brown, M. J., Wishart, H. A., Flashman, L. A., & Santulli, R. B. (2009). Differential memory test sensitivity for diagnosing amnestic mild cognitive impairment and predicting conversion to Alzheimer's disease. Aging, Neuropsychology, and Cognition, 16(3), 357-376.

Drew M. Altschul, Ph.D., is a research psychologist at the University of Edinburgh. His current work lies at the intersection of quantitative psychology, epidemiology, and public health.

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Table of Contents

The problem-solving process, how to solve problems: 5 steps, train to solve problems with lean today, what is problem solving steps, techniques, & best practices explained.

Problem solving is the art of identifying problems and implementing the best possible solutions. Revisiting your problem-solving skills may be the missing piece to leveraging the performance of your business, achieving Lean success, or unlocking your professional potential. 

Ask any colleague if they’re an effective problem-solver and their likely answer will be, “Of course! I solve problems every day.” 

Problem solving is part of most job descriptions, sure. But not everyone can do it consistently. 

Problem solving is the process of defining a problem, identifying its root cause, prioritizing and selecting potential solutions, and implementing the chosen solution.

There’s no one-size-fits-all problem-solving process. Often, it’s a unique methodology that aligns your short- and long-term objectives with the resources at your disposal. Nonetheless, many paradigms center problem solving as a pathway for achieving one’s goals faster and smarter. 

One example is the Six Sigma framework , which emphasizes eliminating errors and refining the customer experience, thereby improving business outcomes. Developed originally by Motorola, the Six Sigma process identifies problems from the perspective of customer satisfaction and improving product delivery. 

Lean management, a similar method, is about streamlining company processes over time so they become “leaner” while producing better outcomes. 

Trendy business management lingo aside, both of these frameworks teach us that investing in your problem solving process for personal and professional arenas will bring better productivity.

1. Precisely Identify Problems

As obvious as it seems, identifying the problem is the first step in the problem-solving process. Pinpointing a problem at the beginning of the process will guide your research, collaboration, and solutions in the right direction. 

At this stage, your task is to identify the scope and substance of the problem. Ask yourself a series of questions: 

• What’s the problem? 
• How many subsets of issues are underneath this problem? 
• What subject areas, departments of work, or functions of business can best define this problem? 

Although some problems are naturally large in scope, precision is key. Write out the problems as statements in planning sheets . Should information or feedback during a later step alter the scope of your problem, revise the statements. 

Framing the problem at this stage will help you stay focused if distractions come up in later stages. Furthermore, how you frame a problem will aid your search for a solution. A strategy of building Lean success, for instance, will emphasize identifying and improving upon inefficient systems. 

2. Collect Information and Plan 

The second step is to collect information and plan the brainstorming process. This is another foundational step to road mapping your problem-solving process. Data, after all, is useful in identifying the scope and substance of your problems. 

Collecting information on the exact details of the problem, however, is done to narrow the brainstorming portion to help you evaluate the outcomes later. Don’t overwhelm yourself with unnecessary information — use the problem statements that you identified in step one as a north star in your research process. 

This stage should also include some planning. Ask yourself:

• What parties will ultimately decide a solution? 
• Whose voices and ideas should be heard in the brainstorming process? 
• What resources are at your disposal for implementing a solution? 

Establish a plan and timeline for steps 3-5. 

3. Brainstorm Solutions

Brainstorming solutions is the bread and butter of the problem-solving process. At this stage, focus on generating creative ideas. As long as the solution directly addresses the problem statements and achieves your goals, don’t immediately rule it out. 

Moreover, solutions are rarely a one-step answer and are more like a roadmap with a set of actions. As you brainstorm ideas, map out these solutions visually and include any relevant factors such as costs involved, action steps, and involved parties. 

With Lean success in mind, stay focused on solutions that minimize waste and improve the flow of business ecosystems. 

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4. Decide and Implement

The most critical stage is selecting a solution. Easier said than done. Consider the criteria that has arisen in previous steps as you decide on a solution that meets your needs. 

Once you select a course of action, implement it. 

Practicing due diligence in earlier stages of the process will ensure that your chosen course of action has been evaluated from all angles. Often, efficient implementation requires us to act correctly and successfully the first time, rather than being hurried and sloppy. Further compilations will create more problems, bringing you back to step 1. 

5. Evaluate

Exercise humility and evaluate your solution honestly. Did you achieve the results you hoped for? What would you do differently next time? 

As some experts note, formulating feedback channels into your evaluation helps solidify future success. A framework like Lean success, for example, will use certain key performance indicators (KPIs) like quality, delivery success, reducing errors, and more. Establish metrics aligned with company goals to assess your solutions.

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SAT II Math I : Solving Functions from Word Problems

Study concepts, example questions & explanations for sat ii math i, all sat ii math i resources, example questions, example question #1 : solving functions from word problems.

At Joe's pizzeria a pizza costs $5 with the first topping, and then an additional 75 cents for each additional topping.

Notice that the question describes a linear equation because there is a constant rate of change (the cost per topping). This means we can use slope intercept form to describe the scenario. 

Recall that slope intercept form is

Putting all these steps together we get:

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Linear Functions Problems with Solutions

Linear functions are highly used throughout mathematics and are therefore important to understand. A set of problems involving linear functions , along with detailed solutions, are presented. The problems are designed with emphasis on the meaning of the slope and the y intercept.

Problem 1: f is a linear function. Values of x and f(x) are given in the table below; complete the table.

Problem 2: A family of linear functions is given by

Solution to Problem 2: a)

Problem 3: A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P ; grows as a linear function of time t, where t is the number of years after 2003. a) How many students will be enrolled in the school in 2010? b) Find a linear function that relates the student population to the time t. Solution to Problem 3: a) The given information may be written as ordered pairs (t , P). The year 2003 correspond to t = 0 and the year 2006 corresponds to t = 3, hence the 2 ordered pairs (0, 1200) and (3, 1500) Since the population grows linearly with the time t, we use the two ordered pairs to find the slope m of the graph of P as follows m = (1500 - 1200) / (6 - 3) = 100 students / year The slope m = 100 means that the students population grows by 100 students every year. From 2003 to 2010 there are 7 years and the students population in 2010 will be P(2010) = P(2003) + 7 * 100 = 1200 + 700 = 1900 students. b) We know the slope and two points, we may use the point slope form to find an equation for the population P as a function of t as follows P - P1 = m (t - t1) P - 1200 = 100 (t - 0) P = 100 t + 1200

Problem 4: The graph shown below is that of the linear function that relates the value V (in $) of a car to its age t, where t is the number of years after 2000.

Problem 5: The cost of producing x tools by a company is given by

Problem 6: A 500-liter tank full of oil is being drained at the constant rate of 20 liters par minute. a) Write a linear function V for the number of liters in the tank after t minutes (assuming that the drainage started at t = 0). b) Find the V and the t intercepts and interpret them. e) How many liters are in the tank after 11 minutes and 45 seconds? Solution to Problem 6: After each minute the amount of oil in the tank deceases by 20 liters. After t minutes, the amount of oil in the tank decreases by 20*t liters. Hence if at the start there 500 liters, after t minute the amount V of oil left in the tank is given by V = 500 - 20 t b) To find the V intercept, set t = 0 in the equation V = 500 - 20 t. V = 500 liters : it is the amount of oil at the start of the drainage. To find the t intercept, set V = 0 in the equation V = 500 - 20 t and solve for t. 0 = 500 - 20 t t = 500 / 20 = 25 minutes : it is the total time it takes to drain the 500 liters of oil. c) Convert 11 minutes 45 seconds in decimal form. t = 11 minutes 45 seconds = 11.75 minutes Calculate V at t = 11.75 minutes. V(11.75) = 500 - 20*11.75 = 265 liters are in the tank after 11 minutes 45 seconds of drainage.

Problem 7: A 50-meter by 70-meter rectangular garden is surrounded by a walkway of constant width x meters.

Problem 8: A driver starts a journey with 25 gallons in the tank of his car. The car burns 5 gallons for every 100 miles. Assuming that the amount of gasoline in the tank decreases linearly, a) write a linear function that relates the number of gallons G left in the tank after a journey of x miles. b) What is the value and meaning of the slope of the graph of G? c) What is the value and meaning of the x intercept? Solution to Problem 8: a) If 5 gallons are burnt for 100 miles then (5 / 100) gallons are burnt for 1 mile. Hence for x miles, x * (5 / 100) gallons are burnt. G is then equal to the initial amount of gasoline decreased by the amount gasoline burnt by the car. Hence G = 25 - (5 / 100) x b) The slope of G is equal to 5 / 1000 and it represent the amount of gasoline burnt for a distance of 1 mile. c) To find the x intercept, we set G = 0 and solve for x. 25 - (5 / 100) x = 0 x = 500 miles : it is the distance x for which all 25 gallons of gasoline will be burnt.

Problem 9: A rectangular wire frame has one of its dimensions moving at the rate of 0.5 cm / second. Its width is constant and equal to 4 cm. If at t = 0 the length of the rectangle is 10 cm,

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What Is Problem Solving?

You will often see beach clean-up drives being publicized in coastal cities. There are already dustbins available on the beaches,…

You will often see beach clean-up drives being publicized in coastal cities. There are already dustbins available on the beaches, so why do people need to organize these drives? It’s evident that despite advertising and posting anti-littering messages, some of us don’t follow the rules.

Temporary food stalls and shops make it even more difficult to keep the beaches clean. Since people can’t ask the shopkeepers to relocate or prevent every single person from littering, the clean-up drive is needed.  This is an ideal example of problem-solving psychology in humans. ( 230-fifth.com ) So, what is problem-solving? Let’s find out.

What Is Problem-Solving?

At its simplest, the meaning of problem-solving is the process of defining a problem, determining its cause, and implementing a solution. The definition of problem-solving is rooted in the fact that as humans, we exert control over our environment through solutions. We move forward in life when we solve problems and make decisions. 

We can better define the problem-solving process through a series of important steps.

Identify The Problem: 

This step isn’t as simple as it sounds. Most times, we mistakenly identify the consequences of a problem rather than the problem itself. It’s important that we’re careful to identify the actual problem and not just its symptoms. 

Define The Problem: 

Once the problem has been identified correctly, you should define it. This step can help clarify what needs to be addressed and for what purpose.

Form A Strategy: 

Develop a strategy to solve your problem. Defining an approach will provide direction and clarity on the next steps. 

Organize The Information:  

Organizing information systematically will help you determine whether something is missing. The more information you have, the easier it’ll become for you to arrive at a solution.  

Allocate Resources:  

We may not always be armed with the necessary resources to solve a problem. Before you commit to implementing a solution for a problem, you should determine the availability of different resources—money, time and other costs.

Track Progress: 

The true meaning of problem-solving is to work towards an objective. If you measure your progress, you can evaluate whether you’re on track. You could revise your strategies if you don’t notice the desired level of progress. 

Evaluate The Results:  

After you spot a solution, evaluate the results to determine whether it’s the best possible solution. For example, you can evaluate the success of a fitness routine after several weeks of exercise.

Meaning Of Problem-Solving Skill

Now that we’ve established the definition of problem-solving psychology in humans, let’s look at how we utilize our problem-solving skills.  These skills help you determine the source of a problem and how to effectively determine the solution. Problem-solving skills aren’t innate and can be mastered over time. Here are some important skills that are beneficial for finding solutions.

Communication

Communication is a critical skill when you have to work in teams.  If you and your colleagues have to work on a project together, you’ll have to collaborate with each other. In case of differences of opinion, you should be able to listen attentively and respond respectfully in order to successfully arrive at a solution.

As a problem-solver, you need to be able to research and identify underlying causes. You should never treat a problem lightly. In-depth study is imperative because often people identify only the symptoms and not the actual problem.

Once you have researched and identified the factors causing a problem, start working towards developing solutions. Your analytical skills can help you differentiate between effective and ineffective solutions.

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Physics > Data Analysis, Statistics and Probability

Title: an inversion problem for optical spectrum data via physics-guided machine learning.

Abstract: We propose the regularized recurrent inference machine (rRIM), a novel machine-learning approach to solve the challenging problem of deriving the pairing glue function from measured optical spectra. The rRIM incorporates physical principles into both training and inference and affords noise robustness, flexibility with out-of-distribution data, and reduced data requirements. It effectively obtains reliable pairing glue functions from experimental optical spectra and yields promising solutions for similar inverse problems of the Fredholm integral equation of the first kind.

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More From Forbes

From hollywood to hr: what business recruiters can learn from casting directors.

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Founder and Chief Culture Officer of Ideal Outcomes , Inc. Author of the new book Culture Ignited: 5 Disciplines for Adaptive Leadership.

If you watched the Oscars ceremony, you saw renowned actors clutching their Academy Awards and making their acceptance speeches. You also likely witnessed a parade of lesser-known creative professionals whose work greatly contributed to the success of their movies.

There were awards for screenplay, music, visual effects, cinematography, film editing, sound, costume design and more. But there’s one pivotal role that wasn’t recognized, though that’s about to change. And that’s for casting directors. Starting with films released in 2025, casting directors will be eligible for an Oscar—the first new award category in more than 20 years, according to the Associated Press .

What has taken so long? I got to thinking about that and how the work that a casting director performs might be instructive in a company’s hiring process. Both functions share the fundamental goal of matching the right individual with the right position to create a successful outcome. It’s much more than fitting skills listed on a resume to those required by the job description. Here’s what I believe HR personnel and senior corporate executives responsible for hiring can learn from their Hollywood counterparts.

Knowing The Position Inside-Out

Casting directors must analyze a script to understand characters’ personalities, motivations and relationships to ensure actors fit these parts. Similarly, HR professionals and recruiters need to have a deep understanding of the job description, company culture and team dynamics to find candidates who not only have the necessary skills but will also thrive in the company environment.

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When a casting director picks an actor, they have to consider whether the individual meets the requirements of the role and has the talent for the job. A business recruiter should similarly scrutinize an applicant’s resume. What jobs has a candidate held that qualify them to take on this new position that, in all likelihood, is a promotion of some kind? Has their career path given them the necessary experience?

Team Player

A movie has an entire cast of characters that could range from the “A list” key players to a horde of extras. On Hollywood sets, having chemistry among the cast can help ensure the film successfully entertains and grips an audience. In the corporate world, it’s just as important to make sure the interviewee is someone who, once hired, will be embraced by the existing team and fits the corporate culture. That means, in the first instance, the recruiter needs to understand the company culture inside-out.

Long-Term Vision

The team casting a movie might need to consider not only how an actor is suited for a role today but also how they may evolve into sequels or series. Similarly, company recruiters should consider a candidate’s long-term potential within the company. Will they be able to advance and grow into higher-level positions? How likely are they to remain on board? By asking questions such as these, recruiters and HR professionals can enhance their talent acquisition strategies, making the process more dynamic and effective in finding winning employees who want to stay with the company.

Creative Problem-Solving

If a casting director is finding it challenging to find the right actor for a part, they might need to think outside the box or consider unconventional choices. HR professionals should learn from this and not get boxed in and hire stereotypes. Through creative sourcing and interviewing techniques, they can uncover hidden talent that brings a fresh approach and breakthrough ideas to a company.

Building Relationships

The world of movie-making is competitive, and knowing a wide array of talent and being able to match that talent with projects quickly is key. Having an extensive database or network within the entertainment industry is a resource that can help casting directors conjure up exceptional talent for a wide variety of roles. Business recruiters can learn from this approach. If interviewees are not the perfect fit for one position but are still strong candidates, recruiters can keep them in mind in case they’re more suited to future opportunities.

The Personal Touch

Inevitably, casting directors have to reject dozens of talented people in favor of the one who gets the role, but they may need to maintain a good relationship for future endeavors. HR professionals should consider this and provide constructive feedback to unsuccessful candidates. This can help turn a potentially negative experience into a positive one and, in the process, maintain a good employer brand.

Final Thoughts

Casting directors and HR professionals use a range of techniques to evaluate their candidates’ or actors’ qualities beyond the basic requirements of the role. For casting directors, this might include screen tests, readings and group auditions, while HR professionals may use behavioral interviews, assessment centers or personality tests. The aim is to uncover the deeper, less visible qualities that contribute to a successful and dynamic team, ensuring that each new addition will enhance the collective performance and output.

The comparison between casting directors’ approach to evaluating actors and recruiters' or HR professionals’ methods for assessing candidates highlights a critical aspect of talent acquisition: the importance of looking beyond surface-level qualifications to identify a truly great match.

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