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Study Guides > ALGEBRA / TRIG I

Evaluating Functions

Learning OUTCOMES

  • Given a function equation, find function values (outputs) for specified numbers and variables (inputs)
Throughout this course, you have been working with algebraic equations. Many of these equations are functions. For example, [latex]y=4x+1[/latex] is an equation that represents a function. When you input values for [latex]x[/latex], you can determine a single output for [latex]y[/latex]. In this case, if you substitute [latex]x=10[/latex] into the equation you will find that [latex]y[/latex] must be [latex]41[/latex]; there is no other value of [latex]y[/latex] that would make the equation true. Rather than using the variable [latex]y[/latex], 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 value in for [latex]x[/latex]. Equations written using function notation can also be evaluated. With function notation, you might see the following: Given [latex]f(x)=4x+1[/latex], find [latex]f(2)[/latex]. You read this problem like this: “given [latex]f[/latex] of [latex]x[/latex] equals [latex]4x[/latex] plus one, find [latex]f[/latex] of [latex]2[/latex].” 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 [latex]2[/latex] for [latex]x[/latex], multiply it by [latex]4[/latex] and add [latex]1[/latex], simplifying to get [latex]9[/latex]. In both a function and an equation, an input of [latex]2[/latex] results in an output of [latex]9[/latex].

[latex]f(x)=4x+1\\f(2)=4(2)+1=8+1=9[/latex]

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

Example

Given [latex]f(x)=3x–4[/latex], find [latex]f(5)[/latex].

Answer: Substitute [latex]5[/latex] in for [latex]x[/latex] in the function.

[latex]f(5)=3(5)-4[/latex]

Simplify the expression on the right side of the equation.

[latex]f(5)=15-4\\f(5)=11[/latex]

Functions can be evaluated for negative values of [latex]x[/latex], too. Keep in mind the rules for integer operations.

Example

Given [latex]p(x)=2x^{2}+5[/latex], find [latex]p(−3)[/latex].

Answer: Substitute [latex]-3[/latex] in for [latex]x[/latex] in the function.

[latex]p(−3)=2(−3)^{2}+5[/latex]

Simplify the expression on the right side of the equation.

[latex]p(−3)=2(9)+5\\p(−3)=18+5\\p(−3)=23[/latex]

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

Example

Given [latex]f(x)=|4x-3|[/latex], find [latex]f(0)[/latex], [latex]f(2)[/latex], and [latex]f(−1)[/latex].

Answer: Treat each of these like three separate problems. In each case, you substitute the value in for [latex]x[/latex] and simplify. Start with [latex]x=0[/latex].

[latex]f(0)=|4(0)-3|=|-3|=3\\f(0)=3[/latex]

Evaluate for [latex]x=2[/latex].

[latex]f(2)=|4(2)-3|=|5|=5\\f(2)=5[/latex]

Evaluate for [latex]x=−1[/latex].

[latex]f(−1)=|4(-1)-3|=|-7|=7\\f(-1)=7[/latex]

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.

Example

Given [latex]f(x)=3x^{2}+2x+1[/latex], find [latex]f(b)[/latex].

Answer: This problem is asking you to evaluate the function for [latex]b[/latex]. This means substitute [latex]b[/latex] in the equation for [latex]x[/latex]. [latex-display]f(b)=3b^{2}+2b+1[/latex-display] (That is all—you are done.)

In the following example, you evaluate a function for an expression. So here you will substitute the entire expression in for [latex]x[/latex] and simplify.

Example

Given [latex]f(x)=4x+1[/latex], find [latex]f(h+1)[/latex].

Answer: This time, you substitute [latex](h+1)[/latex] into the equation for [latex]x[/latex]. [latex]f(h+1)=4(h+1)+1[/latex]  Use the distributive property on the right side, and then combine like terms to simplify. [latex-display]f(h+1)=4h+4+1=4h+5[/latex-display] Given [latex]f(x)=4x+1[/latex], [latex]f(h+1)=4h+5[/latex].

In the following video, we show more examples of evaluating functions for both integer and variable inputs. https://youtu.be/_bi0B2zibOg

Summary

Function notation takes the form such as [latex]f(x)=18x–10[/latex] and is read “f of x equals 18 times x minus [latex]10[/latex].” Function notation can use letters other than [latex]f[/latex], such as [latex]c(x)[/latex], [latex]g(x)[/latex], or [latex]h(x)[/latex]. 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 [latex]x[/latex] and [latex]y[/latex] can be evaluated for different values of the input [latex]x[/latex], an equation written in function notation can also be evaluated for different values of [latex]x[/latex]. To evaluate a function, substitute in values for [latex]x[/latex] and simplify to find the related output.

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  • Ex: Determine Various Function Outputs for a Quadratic Function. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
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