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Evaluating Functions

Author: Sophia
PDF Version

what's covered
In this lesson, you will learn how to evaluate a function for a given value. Specifically, this lesson will cover:

Table of Contents

1. Review of Function Notation

Before we go into evaluating a function for a given value, let's first review the basics of function notation. Recall that the notation f open parentheses x close parentheses is read, "f of x". The meaning behind this notation is that we are evaluating the function for some x value. We often times refer to x as the argument for the function f.

hint
Keep in mind that f open parentheses x close parentheses does not mean f multiplied by x. This is an easy mistake to make.

term to know
Argument
The input value of a function, on which the value of a function depends.

2. Evaluating f  (a  )

Now that we have reviewed function notation, let's take a look at what it means to evaluate a function for some value of x.

EXAMPLE

Suppose we have the equation f open parentheses x close parentheses equals 2 x plus 4 and are asked to evaluate this function when x equals 5. What steps should we take?

First, we need to realize that when we are being asked to evaluate a function, what we are really being asked to do is determine the function's value when the variable that it is dependent upon is equal to a certain value. In this case, the function f open parentheses x close parentheses is dependent upon x, and we want to know the value of f open parentheses 5 close parentheses or f open parentheses x equals 5 close parentheses. To evaluate the function for when x equals 5, we simply replace each x variable with 5, as shown below:

f open parentheses x close parentheses equals 2 x plus 4 Substitute 5 in for x
f open parentheses 5 close parentheses equals 2 open parentheses 5 close parentheses plus 4 Multiply 2 and 5
f open parentheses 5 close parentheses equals 10 plus 4 Add 10 and 4
f open parentheses 5 close parentheses equals 14 Our solution

3. Finding f  (a  ) on a Graph

Sometimes you may be given the graph of a function and asked to determine some value of the function f open parentheses a close parentheses. When you are asked to do this, the process is very similar to if you were given the graph of the equation and asked to find the value of y for a specific value of x.

EXAMPLE

In the equation y equals x squared plus 4, find the value of y when x equals 1. In this case, you would simply locate the point on the coordinate plane where x equals 1 and then move vertically up or down until you touch a point on the graph for the line y equals x squared plus 4. Next, you move horizontally towards the y-axis and identify the point where you touch the y-axis. The y-value of the coordinate system at this point represents the y in this equation.



We can rewrite the previous equation with f open parentheses x close parentheses instead of y to get f open parentheses x close parentheses equals x squared plus 4. If we are asked to find the value of the function when x equals 1, we follow the same process we did before when x equals 1. The only difference is that this time we are saying that f open parentheses 1 close parentheses is equal to some value instead of y. We would write this as f open parentheses 1 close parentheses equals 5.

4. Evaluating f  (x + a  )

Sometimes we may be asked to evaluate a function for some value given in the form f open parentheses x plus a close parentheses. The process we follow is the same as we did for a single argument, only now instead of replacing the x's with a's we replace them with x plus a.

EXAMPLE

Suppose we are given the function f open parentheses x close parentheses equals 5 x minus 2 and asked to evaluate the function for f open parentheses x plus 2 close parentheses. How would we evaluate this function?

f open parentheses x close parentheses equals 5 x minus 2 Substitute open parentheses x plus 2 close parentheses in for x
f open parentheses x plus 2 close parentheses equals 5 open parentheses x plus 2 close parentheses minus 2 Distribute 5 into open parentheses x plus 2 close parentheses
f open parentheses x plus 2 close parentheses equals 5 x plus 10 minus 2 Combine like terms
f open parentheses x plus 2 close parentheses equals 5 x plus 8 Our solution

hint
Whenever evaluating a function for some value, simply replace the variable of the function with the quantity you want to evaluate it for and then simplify.

summary
Reviewing function notation, recall that f open parentheses x close parentheses is read "f of x". The function f open parentheses x close parentheses is a function of x, where x is referred to as the argument. The argument of the function is the input value on which the value of the function depends. You can evaluate a function algebraically by substituting a given value in the domain for x, or evaluate f  (a  ) where a is a given value. You can evaluate a function graphically by finding f open parentheses a close parentheses on a graph. To do this, find a given value in the domain on the x-axis and determine the corresponding y-value of the function. Sometimes you are asked to evaluate f  (x + a  ). The process you follow is the same as you did for a single argument, only now instead of replacing the x's with a's, you replace them with (x + a  ).

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Argument

The input value of a function, on which the value of a function depends.

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