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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, y=4x+1y=4x+1 is an equation that represents a function. When you input values for xx, you can determine a single output for yy. In this case, if you substitute x=10x=10 into the equation you will find that yy must be 4141; there is no other value of yy that would make the equation true. Rather than using the variable yy, 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 xx. Equations written using function notation can also be evaluated. With function notation, you might see the following: Given f(x)=4x+1f(x)=4x+1, find f(2)f(2). You read this problem like this: “given ff of xx equals 4x4x plus one, find ff of 22.” 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 22 for xx, multiply it by 44 and add 11, simplifying to get 99. In both a function and an equation, an input of 22 results in an output of 99.

f(x)=4x+1f(2)=4(2)+1=8+1=9f(x)=4x+1\\f(2)=4(2)+1=8+1=9

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 f(x)f(x) does not mean ff multiplied by xx. Instead, the notation means “f of x” or “the function of xx." To evaluate the function, take the value given for xx, and substitute that value in for xx in the expression. Let us look at a couple of examples.

Example

Given f(x)=3x4f(x)=3x–4, find f(5)f(5).

Answer: Substitute 55 in for xx in the function.

f(5)=3(5)4f(5)=3(5)-4

Simplify the expression on the right side of the equation.

f(5)=154f(5)=11f(5)=15-4\\f(5)=11

Functions can be evaluated for negative values of xx, too. Keep in mind the rules for integer operations.

Example

Given p(x)=2x2+5p(x)=2x^{2}+5, find p(3)p(−3).

Answer: Substitute 3-3 in for xx in the function.

p(3)=2(3)2+5p(−3)=2(−3)^{2}+5

Simplify the expression on the right side of the equation.

p(3)=2(9)+5p(3)=18+5p(3)=23p(−3)=2(9)+5\\p(−3)=18+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.

Example

Given f(x)=4x3f(x)=|4x-3|, find f(0)f(0), f(2)f(2), and f(1)f(−1).

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

f(0)=4(0)3=3=3f(0)=3f(0)=|4(0)-3|=|-3|=3\\f(0)=3

Evaluate for x=2x=2.

f(2)=4(2)3=5=5f(2)=5f(2)=|4(2)-3|=|5|=5\\f(2)=5

Evaluate for x=1x=−1.

f(1)=4(1)3=7=7f(1)=7f(−1)=|4(-1)-3|=|-7|=7\\f(-1)=7

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 f(x)=3x2+2x+1f(x)=3x^{2}+2x+1, find f(b)f(b).

Answer: This problem is asking you to evaluate the function for bb. This means substitute bb in the equation for xx. f(b)=3b2+2b+1f(b)=3b^{2}+2b+1 (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 xx and simplify.

Example

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

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

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 f(x)=18x10f(x)=18x–10 and is read “f of x equals 18 times x minus 1010.” Function notation can use letters other than ff, such as c(x)c(x), g(x)g(x), or h(x)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 xx and yy can be evaluated for different values of the input xx, an equation written in function notation can also be evaluated for different values of xx. To evaluate a function, substitute in values for xx and simplify to find the related output.

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