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Algebra 1 : How to find f(x)

Study concepts, example questions & explanations for algebra 1, all algebra 1 resources, example questions, example question #1 : how to find f(x).

how to solve f(x) math problems

Example Question #2 : How To Find F(X)

how to solve f(x) math problems

Example Question #3 : How To Find F(X)

how to solve f(x) math problems

Example Question #5 : How To Find F(X)

What is the next number in the following sequence?

how to solve f(x) math problems

To form this sequence, alternately multiply by 2 and add 5:

how to solve f(x) math problems

To keep the pattern going, double the seventh term to get the eighth:

how to solve f(x) math problems

Example Question #8 : How To Find F(X)

What is the next number in the following sequence:

how to solve f(x) math problems

To get each member of this sequence, add a number that increases by one with each element:

how to solve f(x) math problems

To get the next element, add 7:

how to solve f(x) math problems

Example Question #9 : How To Find F(X)

how to solve f(x) math problems

Example Question #10 : How To Find F(X)

how to solve f(x) math problems

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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].

Graph of a parabola with labeled points (-3, 3), (1, 3), and (4, 24).

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].

Graph of a positive parabola centered at (1, 0).

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]

how to solve f(x) math problems

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how to solve f(x) math problems

How Do You Find f(x) If You Have a Value For x?

To solve a function for a given value, plug that value into the function and simplify. See this first-hand by watching this tutorial!

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Function definitions and notation.

What is Function Notation?

What is Function Notation?

Every see 'f(x)' in your math? That's function notation! It's a way to indicate that an equation is a function. Learn about function notation by watching this tutorial.

Further Exploration

How Do You Solve a Word Problem Using a Function?

How Do You Solve a Word Problem Using a Function?

Word problems are a great way to see math in action! This word problem deals with calculating profit after a certain number of years. See how to use a function from the word problem to solve!

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How to Solve for X

Last Updated: March 20, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 797,207 times.

There are a number of ways to solve for x, whether you're working with exponents and radicals or if you just have to do some division or multiplication. No matter what process you use, you always have to find a way to isolate x on one side of the equation so you can find its value. Here's how to do it:

Using a Basic Linear Equation

Step 1 Write down the problem.

  • 2 2 (x+3) + 9 - 5 = 32

Step 2 Resolve the exponent.

  • 4(x+3) + 9 - 5 = 32

Step 3 Do the multiplication....

  • 4x + 12 + 9 - 5 = 32

Step 4 Do the addition...

  • 4x+21-5 = 32
  • 4x + 16 - 16 = 32 - 16

Step 5 Isolate the variable.

  • 4x/4 = 16/4

Step 6 Check your work.

  • 2 2 (x+3)+ 9 - 5 = 32
  • 2 2 (4+3)+ 9 - 5 = 32
  • 2 2 (7) + 9 - 5 = 32
  • 4(7) + 9 - 5 = 32
  • 28 + 9 - 5 = 32
  • 37 - 5 = 32

With Exponents

Step 1 Write down the problem.

  • 2x 2 + 12 = 44

Step 2 Isolate the term with the exponent.

  • 2x 2 +12-12 = 44-12

Step 3 Isolate the variable with the exponent by dividing both sides by the coefficient of the x term.

  • (2x 2 )/2 = 32/2
  • 4 Take the square root of each side of the equation. [6] X Research source Taking the square root of x 2 will cancel it out. So, take the square root of both sides. You'll get x left over on one side and plus or minus the square root of 16, 4, on the other side. Therefore, x = ±4.
  • 2(4) 2 + 12 = 44
  • 2(16) + 12 = 44
  • 32 + 12 = 44

Using Fractions

Step 1 Write down the problem.

  • (x + 3)/6 = 2/3

Step 2 Cross multiply...

  • (x + 3) x 3 = 3x + 9
  • 3x + 9 = 12

Step 3 Combine like terms.

  • 3x + 9 - 9 = 12 - 9

Step 4 Isolate x by dividing each term by the x coefficient.

  • (1 + 3)/6 = 2/3

Using Radical Signs

Step 1 Write down the problem.

  • √(2x+9) - 5 = 0

Step 2 Isolate the square root.

  • √(2x+9) - 5 + 5 = 0 + 5
  • √(2x+9) = 5

Step 3 Square...

  • (√(2x+9)) 2 = 5 2
  • 2x + 9 = 25

Step 4 Combine like terms.

  • 2x + 9 - 9 = 25 - 9

Step 5 Isolate the variable.

  • √(2(8)+9) - 5 = 0
  • √(16+9) - 5 = 0
  • √(25) - 5 = 0

Using Absolute Value

Step 1 Write down the problem.

  • |4x +2| - 6 = 8

Step 2 Isolate the absolute...

  • |4x +2| - 6 + 6 = 8 + 6
  • |4x +2| = 14

Step 3 Remove the absolute value and solve the equation.

  • 4x + 2 = 14
  • 4x + 2 - 2 = 14 -2

Step 4 Remove the absolute value and change the sign of the terms on the opposite side of the equal sign before you solve.

  • 4x + 2 = -14
  • 4x + 2 - 2 = -14 - 2
  • 4x/4 = -16/4

Step 5 Check your work.

  • |4(3) +2| - 6 = 8
  • |12 +2| - 6 = 8
  • |14| - 6 = 8
  • |4(-4) +2| - 6 = 8
  • |-16 +2| - 6 = 8
  • |-14| - 6 = 8

Practice Problems and Answers

how to solve f(x) math problems

Expert Q&A

David Jia

  • To check your work, plug the value of x back into the original equation and solve. Thanks Helpful 0 Not Helpful 0
  • Radicals, or roots, are another way of representing exponents. The square root of x = x^1/2. Thanks Helpful 0 Not Helpful 0

how to solve f(x) math problems

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Find the Maximum or Minimum Value of a Quadratic Function Easily

  • ↑ David Jia. Academic Tutor. Expert Interview. 23 February 2021
  • ↑ https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearEqns.aspx
  • ↑ https://www.purplemath.com/modules/solvelin.htm
  • ↑ https://sciencing.com/tips-for-solving-algebraic-equations-13712207.html
  • ↑ https://www.youtube.com/watch?v=LMS1NR4gZN8
  • ↑ https://www.mathsisfun.com/algebra/fractions-algebra.html
  • ↑ http://www.mathsisfun.com/algebra/radical-equations-solving.html
  • ↑ http://www.sosmath.com/algebra/solve/solve0/solve0.html

About This Article

David Jia

To solve for x in a basic linear equation, start by resolving the exponent using the order of operations. Then, isolate the variable to get your answer. To solve for x when the equation includes an exponent, start by isolating the term with the exponent. Then, isolate the variable with the exponent by dividing both sides by the coefficient of the x term to get your answer. If the equation has fractions, start by cross-multiplying the fractions. Then, combine like terms and isolate x by dividing each term by the x coefficient. If you want to learn how to solve for x if the equation has radicals or absolute values, keep reading the article! Did this summary help you? Yes No

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The Sunday Read: ‘Sure, It Won an Oscar. But Is It Criterion?’

How the criterion collection became the film world’s arbiter of taste..

Narrated by Shaun Taylor-Corbett

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In October 2022, amid a flurry of media appearances promoting their film “Tàr,” the director Todd Field and the star Cate Blanchett made time to visit a cramped closet in Manhattan. This closet, which has become a sacred space for movie buffs, was once a disused bathroom at the headquarters of the Criterion Collection, a 40-year-old company dedicated to “gathering the greatest films from around the world” and making high-quality editions available to the public on DVD and Blu-ray and, more recently, through its streaming service, the Criterion Channel. Today Criterion uses the closet as its stockroom, housing films by some 600 directors from more than 50 countries — a catalog so synonymous with cinematic achievement that it has come to function as a kind of film Hall of Fame. Through a combination of luck, obsession and good taste, this 55-person company has become the arbiter of what makes a great movie, more so than any Hollywood studio or awards ceremony.

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Additional production for The Sunday Read was contributed by Isabella Anderson, Anna Diamond, Sarah Diamond, Elena Hecht, Emma Kehlbeck, Tanya Pérez and Krish Seenivasan.

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COMMENTS

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    Explanation: . In the relation , there are many values of that can be paired with more than one value of - for example, . To demonstrate that is a function of in the other examples, we solve each for : can be rewritten as . can be rewritten as can be rewritten as need not be rewritten.

  2. Worked example: Evaluating functions from equation

    1. Yes. In mathematics it is more common to use a single letter (sometimes a Greek letter), but a function name can be anything. After all it's just a way to communicate to other humans what you're talking about, changing a name doesn't change the math. 2. Yes. A simple example is f (x,y) = x * y. 3. Yes.

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    QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and ...

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    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 f(x) = 5 − 3x2 can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

  5. Evaluate functions

    Course: Algebra 1 > Unit 8. Lesson 1: Evaluating functions. What is a function? Worked example: Evaluating functions from equation. Evaluate functions. Worked example: Evaluating functions from graph. Evaluating discrete functions. Evaluate functions from their graph. Worked example: evaluating expressions with function notation.

  6. Functions

    Unit 6 Systems of equations. Unit 7 Inequalities (systems & graphs) Unit 8 Functions. Unit 9 Sequences. Unit 10 Absolute value & piecewise functions. Unit 11 Exponents & radicals. Unit 12 Exponential growth & decay. Unit 13 Quadratics: Multiplying & factoring. Unit 14 Quadratic functions & equations.

  7. Calculus I

    The difference quotient of a function f (x) f ( x) is defined to be, f (x+h) −f (x) h f ( x + h) − f ( x) h. For problems 5 - 9 compute the difference quotient of the given function. f (x) = 4x −9 f ( x) = 4 x − 9 Solution. g(x) = 6−x2 g ( x) = 6 − x 2 Solution. f (t) = 2t2 −3t+9 f ( t) = 2 t 2 − 3 t + 9 Solution.

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  14. How Do You Find f(x) If You Have a Value For x?

    Word problems are a great way to see math in action! This word problem deals with calculating profit after a certain number of years. See how to use a function from the word problem to solve! Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 ...

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  16. 6 Ways to Solve for X

    5. Isolate the variable. The last thing you have to do to solve for x is to isolate the variable by dividing both sides of the equation by 2, the coefficient of the x term. 2x/2 = x and 16/2 = 8, so you're left with x = 8. 6. Check your work. Plug 8 back in to the equation for x to see if you get the right answer:

  17. Finding inverse functions (article)

    Google Classroom. Learn how to find the formula of the inverse function of a given function. For example, find the inverse of f (x)=3x+2. Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if f takes a to b , then the inverse, f − 1 , must take b to a . Or in other words, f ( a) = b f − 1 ( b ...

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  21. How to solve this nice math algebra equation

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  24. The Sunday Read: 'Sure, It Won an Oscar. But Is It Criterion?'

    Listen and follow The Daily Apple Podcasts | Spotify. In October 2022, amid a flurry of media appearances promoting their film "Tàr," the director Todd Field and the star Cate Blanchett made ...

  25. Finding composite functions (video)

    2) f(x)/1 + f(x) = 2*f(x), so it is not equal to the original problem. 3) If you are asking can you divide out the f(x)'s in f(x)/[1+f(x)] to create 1/(1+1), the answer is no. When we reduce fractions, we divide out common factors (items being multiplied). The denominator here has terms. We can't divide out individual terms. Hope this helps.

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