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Course: College Algebra > Unit 11

Lesson 3: Composing functions

Composing functions

Walk through examples, explanations, and practice problems to learn how to find and evaluate composite functions.

Given two functions, we can combine them in such a way so that the outputs of one function become the inputs of the other. This action defines a composite function. Let's take a look at what this means!

Evaluating composite functions

Example

f (x) = 3 x - 1

and

g (x) = x^{3} + 2

, then what is

f (g (3))

Solution

One way to evaluate

f (g (3))

is to work from the "inside out". In other words, let's evaluate

g (3)

first and then substitute that result into

f

to find our answer.

Let's evaluate

g (3)

$\begin{aligned} g (x) & = x^{3} + 2 \\ g (3) & = (3)^{3} + 2 Plug in x = 3. \\ = 29 \end{aligned}$ ‍

Since

g (3) = 29

, then

f (g (3)) = f (29)

Now let's evaluate

f (29)

$\begin{aligned} f (x) & = 3 x - 1 \\ f (29) & = 3 (29) - 1 Plug in x = 29. \\ = 86 \end{aligned}$ ‍

It follows that

f (g (3)) = f (29) = 86

Finding the composite function

In the above example, function

g

took

3

29

, and then function

f

took

29

86

. Let's find the function that takes

3

directly to

86

To do this, we must compose the two functions and find

f (g (x))

Example

What is

f (g (x))

?
For reference, remember that $f (x) = 3 x - 1$ ‍ and $g (x) = x^{3} + 2$ ‍.

Solution

If we look at the expression

f (g (x))

, we can see that

g (x)

is the input of function

f

. So, let's substitute

g (x)

everywhere we see

x

in function

f

$\begin{aligned} f (x) & = 3 x - 1 \\ f (g (x)) & = 3 (g (x)) - 1 \end{aligned}$ ‍

Since

g (x) = x^{3} + 2

, we can substitute

x^{3} + 2

in for

g (x)

$\begin{aligned} f (g (x)) & = 3 (g (x)) - 1 \\ = 3 (x^{3} + 2) - 1 \\ = 3 x^{3} + 6 - 1 \\ = 3 x^{3} + 5 \end{aligned}$ ‍

This new function should take

3

directly to

86

. Let's verify this.

$\begin{aligned} f (g (x)) & = 3 x^{3} + 5 \\ f (g (3)) & = 3 (3)^{3} + 5 \\ = 86 \end{aligned}$ ‍

Excellent!

Let's practice

Problem 1

f (x) = 3 x - 1

g (x) = x^{3} + 2

Evaluate $g (f (1))$ ‍.

To evaluate

g (f (1))

, let's work from the "inside out".

Let's evaluate

f (1)

$\begin{aligned} f (x) & = 3 x - 1 \\ f (1) & = 3 (1) - 1 Plug in x = 1. \\ = 2 \end{aligned}$ ‍

Since

f (1) = 2

, then

g (f (1)) = g (2)

Now let's evaluate

g (2)

$\begin{aligned} g (x) & = x^{3} + 2 \\ g (2) & = (2)^{3} + 2 Plug in x = 2. \\ = 10 \end{aligned}$ ‍

So it follows that

g (f (1)) = g (2) = 10

We could also find

g (f (x))

and evaluate this for

x = 1

Problem 2

m (x) = 3 x - 2

n (x) = x + 4

Find $m (n (x))$ ‍.

Composite functions: a formal definition

In the above example, we found and evaluated a composite function.

In general, to indicate function

f

composed with function

g

, we can write

f \circ g

, read as "

f

composed with

g

". This composition is defined by the following rule:

$(f \circ g) (x) = f (g (x))$ ‍

The diagram below shows the relationship between

(f \circ g) (x)

and

f (g (x))

Now let's look at another example with this new definition in mind.

Example

g (x) = x + 4

h (x) = x^{2} - 2 x

Find

(h \circ g) (x)

and

(h \circ g) (- 2)

Solution

We can find

(h \circ g) (x)

as follows:

$\begin{aligned} (h \circ g) (x) & = h (g (x)) & Define. \\ = (g (x))^{2} - 2 (g (x)) & Plug g (x) in for x in function h . \\ = (x + 4)^{2} - 2 (x + 4) & Substitute x + 4 for g (x) . \\ = x^{2} + 8 x + 16 - 2 x - 8 & Distribute. \\ = x^{2} + 6 x + 8 & Combine like terms. \end{aligned}$ ‍

Since we now have function

h \circ g

, we can simply substitute

- 2

in for

x

to find

(h \circ g) (- 2)

$\begin{aligned} (h \circ g) (x) & = x^{2} + 6 x + 8 \\ (h \circ g) (- 2) & = (- 2)^{2} + 6 (- 2) + 8 \\ = 4 - 12 + 8 \\ = 0 \end{aligned}$ ‍

Of course, we could have also found

(h \circ g) (- 2)

by evaluating

h (g (- 2))

. This is shown below:

$\begin{aligned} (h \circ g) (- 2) & = h (g (- 2)) \\ = h (2) Since g (- 2) = - 2 + 4 = 2 \\ = 0 Since h (2) = 2^{2} - 2 (2) = 0 \end{aligned}$ ‍

The diagram below shows how

(h \circ g) (- 2)

is related to

h (g (- 2))

Here we can see that function

g

takes

- 2

2

and then function

h

takes

2

0

, while function

h \circ g

takes

- 2

directly to

0

Now let's practice some problems

Problem 3

f (x) = 3 x - 5

g (x) = 3 - 2 x

Evaluate $(g \circ f) (3)$ ‍.

First, let's use the definition of function composition to rewrite the problem as

g (f (3))

To evaluate

g (f (3))

, let's work from the "inside out".

Let's evaluate

f (3)

$\begin{aligned} f (x) & = 3 x - 5 \\ f (3) & = 3 (3) - 5 Plug in x = 3. \\ = 4 \end{aligned}$ ‍

Since

f (3) = 4

, then

g (f (3)) = g (4)

Now let's find

g (4)

$\begin{aligned} g (x) & = 3 - 2 x \\ g (4) & = 3 - 2 (4) Plug in x = 4. \\ = - 5 \end{aligned}$ ‍

So it follows that

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

We could also find

g (f (x))

and evaluate this for

x = 3

In problems 4 and 5, let

f (t) = t - 2

and

g (t) = t^{2} + 5

Problem 4

Find $(g \circ f) (t)$ ‍.

Using the definition of function composition, we can first rewrite

(g \circ f) (t)

g (f (t))

Now, if we look at the expression

g (f (t))

, we can see that

f (t)

is the input of function

g

. So, let's substitute

f (t)

everywhere we see

t

in function

g

$\begin{aligned} g (t) & = t^{2} + 5 \\ g (f (t)) & = (f (t))^{2} + 5 \end{aligned}$ ‍

Since

f (t) = t - 2

, we can substitute

t - 2

in for

f (t)

$\begin{aligned} g (f (t)) & = (f (t))^{2} + 5 \\ = (t - 2)^{2} + 5 \\ = t^{2} - 4 t + 4 + 5 \\ = t^{2} - 4 t + 9 \end{aligned}$ ‍

Problem 5

Find $(f \circ g) (t)$ ‍.

Using the definition of function composition, we can first rewrite

(f \circ g) (t)

f (g (t))

Now, if we look at the expression

f (g (t))

, we can see that

g (t)

is the input of function

f

. So, let's substitute

g (t)

everywhere we see

t

in function

f

$\begin{aligned} f (t) & = t - 2 \\ f (g (t)) & = (g (t)) - 2 \end{aligned}$ ‍

Since

g (t) = t^{2} + 5

, we can substitute

t^{2} + 5

in for

g (t)

$\begin{aligned} f (g (t)) & = (g (t)) - 2 \\ = (t^{2} + 5) - 2 \\ = t^{2} + 5 - 2 \\ = t^{2} + 3 \end{aligned}$ ‍

Challenge Problem

The graphs of the equations

y = f (x)

and

y = g (x)

are shown in the grid below.

Which of the following best approximates the value of $(f \circ g) (8)$ ‍?