# Evaluating Functions

### Learning OUTCOMES

- Given a function equation, find function values (outputs) for specified numbers and variables (inputs)

**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]

### 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]

### 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.)

### 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].

## 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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