Use Sophia to knock out your gen-ed requirements quickly and affordably. Learn more
×

Function of a Function

Author: Sophia

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

Table of Contents

1. Composite Functions

You may have noticed that when we evaluate a function, we typically evaluate it for a number or variable, but we can go one step further and evaluate a function of another function. This process is very similar to when you would substitute a variable with an expression.

EXAMPLE

Substitute y equals 3 x plus 1 into 2 x minus y equals 4 and solve for x.

Since y equals 3 x plus 1, we can substitute 3 x plus 1 in for y into 2 x minus y equals 4.

2 x minus y equals 4 Substitute 3 x plus 1 in for y
2 x minus open parentheses 3 x plus 1 close parentheses equals 4 Distribute subtraction
2 x minus 3 x minus 1 equals 4 Combine like terms
short dash x minus 1 equals 4 Add 1 to both sides
short dash x equals 5 Multiply both sides by -1
x equals short dash 5 Our Solution

When working with composite functions, we follow a similar process. If we are given two functions f open parentheses x close parentheses and g open parentheses x close parentheses and we want to find f open parentheses g open parentheses x close parentheses close parentheses we simply replace all instances of x in the function f open parentheses x close parentheses with whatever g open parentheses x close parentheses is set equal to. Then we simplify.

big idea
When we have a composite function, the notation f open parentheses g open parentheses x close parentheses close parentheses can be written as left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis. This expression is read, "f of g of x" and means that we are replacing the x values of inputs of f open parentheses x close parentheses with another function, in this case, g open parentheses x close parentheses.

term to know
Composite Function
The combination of functions, such that the output of one function is the input of another.


2. Evaluating a Composite Function

Let's take a look at how to actually evaluate a composite function.

EXAMPLE

Suppose f left parenthesis x right parenthesis equals x squared minus 1 and g left parenthesis x right parenthesis equals x minus 1. Find f left parenthesis g left parenthesis x right parenthesis right parenthesis.

The composite function f left parenthesis g left parenthesis x right parenthesis right parenthesis means we will replace all instances of x in f open parentheses x close parentheses with the function g open parentheses x close parentheses comma which is g open parentheses x close parentheses equals x minus 1. To do this we can take the following steps:

f open parentheses x close parentheses equals x squared minus 1 Replace x with g open parentheses x close parentheses
f open parentheses g open parentheses x close parentheses close parentheses equals open parentheses g open parentheses x close parentheses close parentheses squared minus 1 Replace g open parentheses x close parentheses with x minus 1
f open parentheses g open parentheses x close parentheses close parentheses equals open parentheses x minus 1 close parentheses squared minus 1 Expand open parentheses x short dash 1 close parentheses squared
f open parentheses g open parentheses x close parentheses close parentheses equals open parentheses x minus 1 close parentheses open parentheses x minus 1 close parentheses minus 1 FOIL
f open parentheses g open parentheses x close parentheses close parentheses equals open parentheses x squared minus 2 x plus 1 close parentheses minus 1 Evaluate
f open parentheses g open parentheses x close parentheses close parentheses equals x squared minus 2 x Our solution

Sometimes we may be asked to solve a particular composite function when x is equal to a given value, for example x equals 2. To make this evaluation, we follow the same steps we did above then at the end we simply substitute the value we are given for x. Let's see how we do this:

EXAMPLE

Using the result from above, evaluate f open parentheses g open parentheses 2 close parentheses close parentheses.

Our result for f open parentheses g open parentheses x close parentheses close parentheses was f open parentheses g open parentheses x close parentheses close parentheses equals x squared minus 2 x. Now substitute 2 in for x.

f open parentheses g open parentheses x close parentheses close parentheses equals x squared minus 2 x Replace x with 2
f open parentheses g open parentheses 2 close parentheses close parentheses equals open parentheses 2 close parentheses squared minus 2 open parentheses 2 close parentheses Evaluate
f open parentheses g open parentheses 2 close parentheses close parentheses equals 4 minus 4 Simplify
f open parentheses g open parentheses 2 close parentheses close parentheses equals 0 Our solution

Alternatively, we could have evaluated g open parentheses 2 close parentheses first and then plug that value in for x in the function f open parentheses x close parentheses. Either method would give us the same result.

g left parenthesis x right parenthesis equals x minus 1 Evaluate g open parentheses 2 close parentheses
g open parentheses 2 close parentheses equals 2 minus 1 Simplify
g open parentheses 2 close parentheses equals 1 Evaluate f open parentheses g open parentheses 2 close parentheses close parentheses by plugging in 1 for g open parentheses 2 close parentheses
f open parentheses g open parentheses 2 close parentheses close parentheses equals f open parentheses 1 close parentheses Evaluate f open parentheses 1 close parentheses
f open parentheses 1 close parentheses equals 1 squared minus 1 Simplify
f open parentheses 1 close parentheses equals 0 Our Solution

hint
Sometimes you may be asked to find g left parenthesis f left parenthesis x right parenthesis right parenthesis. In this case, we do the same procedures only we replace each variable in g left parenthesis x right parenthesis with the expression for f left parenthesis x right parenthesis.


3. Evaluating f  (f  (x  ))

Sometimes you may come across a situation where you will need to evaluate a function for itself, namely f left parenthesis f left parenthesis x right parenthesis right parenthesis. This simply means that you are plugging in the expression of the function for each x in the function.

EXAMPLE

Find f left parenthesis f left parenthesis x right parenthesis right parenthesis for f left parenthesis x right parenthesis equals 2 x minus 1.

f open parentheses x close parentheses equals 2 x minus 1 Replace x with f open parentheses x close parentheses
f open parentheses f open parentheses x close parentheses close parentheses equals 2 open parentheses f open parentheses x close parentheses close parentheses minus 1 Replace f open parentheses x close parentheses with 2 x minus 1 on the right side
f open parentheses f open parentheses x close parentheses close parentheses equals 2 open parentheses 2 x minus 1 close parentheses minus 1 Distribute 2 into 2 x minus 1
f open parentheses f open parentheses x close parentheses close parentheses equals 4 x minus 2 minus 1 Simplify
f open parentheses f open parentheses x close parentheses close parentheses equals 4 x minus 3 Our solution

Like with the previous problem, you may be asked to find the composite function of itself when x is a given value, for example f left parenthesis f left parenthesis 0 right parenthesis right parenthesis. Just like before we simply need to replace each instance of x in the function f left parenthesis f left parenthesis x right parenthesis right parenthesis with the given value.

EXAMPLE

Using the result from above, evaluatef open parentheses f open parentheses 0 close parentheses close parentheses.

f open parentheses f open parentheses x close parentheses close parentheses equals 4 x minus 3 Replace x with 0
f open parentheses f open parentheses 0 close parentheses close parentheses equals 4 open parentheses 0 close parentheses minus 3 Evaluate
f open parentheses f open parentheses 0 close parentheses close parentheses equals 0 minus 3 Simplify
f open parentheses f open parentheses 0 close parentheses close parentheses equals short dash 3 Our solution

We can show this in the graph below. First note that if f left parenthesis x right parenthesis equals 2 x minus 1, then f left parenthesis 0 right parenthesis equals short dash 1. Plugging in -1 in for x in f left parenthesis x right parenthesis we get f left parenthesis short dash 1 right parenthesis equals short dash 3, which is the same answer we found doing this algebraically.

summary
A composite function is a function of a function. We used the notation open parentheses f ring operator g close parentheses open parentheses x close parentheses comma which is read as "f of g of x". When evaluating a composite function, the output of the innermost function becomes the input of the outermost function. It is not always true that open parentheses f ring operator g close parentheses open parentheses x close parentheses is equal to open parentheses g ring operator f close parentheses open parentheses x close parentheses comma Evaluating f  (f  (x  )) simply means that we are plugging in the expression of the function for each x in the function.

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
Composite Function

The combination of functions, such that the output of one function is the input of another.