Topic 2.7A

Composition of Functions

Nova earns $0.02 per subscriber per day from ads. Its owner wants to know: how much does the channel earn per day, 8 weeks after launch? No single function answers that. You need three: one to turn weeks into months, one to turn months into subscribers, and one to turn subscribers into dollars. Feeding the output of one function into the next is called composition.

8 MIN READ7 IDEAS33 PROBLEMS10 flashcards

Practice 33 problems Next: Chapter 2.7B Composition of Functions

Remember from 2.4: we rewrote $N$ with time in weeks as $256(1.5)^{w/4}$ . That was already a composition: first convert $w$ to months, then apply $N$ .

Output In, Output Out

Each function's output becomes the next function's input.

With $T(w) = w/4$ (weeks → months), $N(t) = 256(1.5)^{t}$ (months → subscribers) and $D(S) = 0.02S$ (subscribers → dollars per day):

01
$T(8) = 8/4 = 2$ . Eight weeks is 2 months.

02
$N(2) = 256(1.5)^{2} = 576$ subscribers.

03
$D(576) = 0.02(576)$ = $11.52 per day. In one expression: $D(N(T(8)))$ = 11.52.

CONCEPT

Composition

The composition $f(g(x))$ means: put $x$ into $g$ , then put the result $g(x)$ into $f$ . It is also written ( $f$ ∘ $g)( x$ ), read “f of $g$ of x”.

Work from the INSIDE out: the function written closest to $x$ acts first, even though it is written second.

For $f(g(x))$ to exist, $x$ must be in the domain of $g$ , and $g(x)$ must be in the domain of $f$ .

(f \circ g)(x) = f\big(g(x)\big)

The small circle ∘ is read “composed with”, so $f$ ∘ $g$ is “f composed with g”. It is NOT a multiplication dot. ( $f$ ∘ g)(x) and $f(g(x))$ mean exactly the same thing; the second form shows the order more clearly, so we'll mostly use it.

In $f(g(x))$ , $g$ is applied first and $f$ second.

From Formulas

Worked example

Let $f(x) = x^{2} - 3$ and $g(x) = 2x + 1$ . Find $f(g(2))$ , $g(f(2))$ , and $f(f(2))$ .

01
$f(g(2))$ : inside first, $g(2) = 2(2) + 1 = 5$ . Then $f(5) = 25 - 3 = 22$ .
02
$g(f(2))$ : inside first, $f(2) = 4 - 3 = 1$ . Then $g(1) = 2(1) + 1 = 3$ .
03
$f(f(2))$ : $f(2) = 1$ , then $f(1) = 1 - 3 = -2$ . A function can be composed with itself.
$f(g(2)) = 22$ but $g(f(2)) = 3$ . Swapping the order changed the answer completely.

KEY RULE

Order matters: in general $f(g(x)) \ne g(f(x))$ .

COMMON MISTAKE

“ $f(g(2))$ means $f(2)$ times $g(2)$ .”

Composition is not multiplication. $f(2) \cdot g(2) = 1 \cdot 5 = 5$ , but $f(g(2)) = 22$ . In $f(g(2))$ , the value $g(2)$ goes INTO $f$ .

“ $f(g(2))$ : do $f$ first, because $f$ is written first.”

Read from the inside out. The function next to the input acts first. Doing $f$ first gives $g(f(2)) = 3$ , a different answer.

Quick check

Let $f(x) = 3x + 1$ and $g(x) = x^2$ . Find $f(g(2))$ and $g(f(2))$ .

From a Table

x	−1	0	1	2	3	4
f(x)	3	5	2	0	4	1
g(x)	2	4	3	−1	1	0

Worked example

Use the table to find $f(g(1))$ , $g(f(1))$ , $f(f(2))$ , and $g(f(0))$ .

01
$f(g(1))$ : $g(1) = 3$ , then $f(3) = 4$ . $g(f(1))$ : $f(1) = 2$ , then $g(2) = -1$ .
02
$f(f(2))$ : $f(2) = 0$ , then $f(0) = 5$ .
03
$g(f(0))$ : $f(0) = 5$ , then we need $g(5)$ . The table has no $x = 5$ , so $g(f(0))$ can't be found from this information.

COMMON MISTAKE

“ $g(f(0))$ : $f(0) = 5$ , and I'll just use the last g-value in the table.”

You may only use values that are actually given. If the middle value isn't an input in the table, the composition can't be evaluated from the table. Say so; don't guess.

Quick check

A table gives $f(1) = 4$ , $f(4) = 2$ , $g(2) = 5$ , and $g(4) = 0$ . What is $g(f(1))$ ?

From a Graph

Two piecewise-linear functions, each defined only for $0 \le x \le 6$ .

Worked example

Use the graphs to find $f(g(1))$ , $g(f(2))$ , $f(g(4))$ , and $g(f(5))$ .

01
$f(g(1))$ : on the graph of $g$ , $g(1) = 3$ . On the graph of $f$ , $f(3) = 3$ . So $f(g(1)) = 3$ .
02
$g(f(2))$ : $f(2) = 5$ , then $g(5) = 3$ . And $f(g(4))$ : $g(4) = 2$ , then $f(2) = 5$ .
03
$g(f(5))$ : $f(5) = -1$ , but $g$ is only defined for $0 \le x \le 6$ . Since −1 is not in the domain of $g$ , $g(f(5))$ is undefined.

REAL-LIFE EXAMPLE

Nova's Daily Earnings

$D(N(T(w)))$ gives Nova's daily ad earnings $w$ weeks after launch. At $w = 20$ weeks: $T(20) = 5$ months, $N(5) = 1,944$ subscribers, $D(1944)$ = $38.88 per day.

Notice that each function answers one question, with its own units, and the composition chains the answers together.

Worked example

Try it yourself. (a) With $f(x) = x^{2} - 3$ and $g(x) = 2x + 1$ , find $f(g(3))$ and $g(f(3))$ . (b) From the table, find $g(f(-1))$ and $f(f(4))$ . (c) From the graphs, find $f(g(6))$ .

Answers: (a) $g(3) = 7$ and $f(7) = 46$ ; $f(3) = 6$ and $g(6) = 13$ . (b) $f(-1) = 3$ and $g(3) = 1$ ; $f(4) = 1$ and $f(1) = 2$ . (c) $g(6) = 4$ and $f(4) = 1$ .

Mixing Formulas, Tables, and Graphs

The inside and outside functions don't have to come in the same form. Just evaluate each one in whatever form it is given.

Worked example

Example. $f(x) = 2x - 1$ , and $g$ is given by the table in Section 3. Find $f(g(2))$ , $g(f(2))$ , and $g(f(0))$ .

01
$f(g(2))$ : from the table, $g(2) = -1$ . Then use the formula: $f(-1) = 2(-1) - 1 = -3$ .
02
$g(f(2))$ : formula first, $f(2) = 2(2) - 1 = 3$ . Then the table: $g(3) = 1$ .
03
$g(f(0))$ : $f(0) = -1$ , and the table gives $g(-1) = 2$ .
A function can also be composed with itself on a graph. Using the graphs in Section 4: $g(g(0)) = g(4) = 2$ , and $f(f(0)) = f(1) = 3$ .

And with Nova's chain of formulas: 4 weeks after launch, $T(4) = 1$ month, $N(1) = 384$ subscribers, $D(384)$ = $7.68 per day.

COMMON MISTAKE

“To find $f(g(1))$ on the graph, I look for where $g$ equals 1.”

That's running $g$ backwards. $f(g(1))$ starts at the INPUT $x = 1$ on the graph of $g$ : go to $x = 1$ , read the height $g(1) = 3$ , then use 3 as the input of $f$ .

Practice

Worked example

P1. $h(x) = 3x - 1$ and $k(x) = x^{2}$ . Find $h(k(2))$ and $k(h(2))$ .

Answer: $k(2) = 4$ , $h(4) = 11$ . $h(2) = 5$ , $k(5) = 25$ . Different, because order matters.

Worked example

P2. Use the table in Section 3 to find $f(g(3))$ .

Answer: $g(3) = 1$ , then $f(1) = 2$ .

Worked example

P3. Find Nova's daily earnings 12 weeks after launch, $D(N(T(12)))$ .

Answer: $T(12) = 3$ months, $N(3) = 864$ subscribers, $D(864)$ = $17.28 per day.

Worked example

P4. With $f(x) = 2x - 1$ and the table in Section 3, find $f(g(4))$ and $g(f(1))$ .

Answer: $g(4) = 0$ , so $f(0) = -1$ . $f(1) = 1$ , so $g(1) = 3$ .

Common slips

“ $f(g(2))$ means $f(2)$ times $g(2)$ .”

Composition is not multiplication. $f(2) \cdot g(2) = 1 \cdot 5 = 5$ , but $f(g(2)) = 22$ . In $f(g(2))$ , the value $g(2)$ Goes into $f$ .

“ $f(g(2))$ : do $f$ first, because $f$ is written first.”

Read from the inside out. The function next to the input acts first. Doing $f$ first gives $g(f(2)) = 3$ , a different answer.
“ $g(f(0))$ : $f(0) = 5$ , and I'll just use the last g-value in the table.”

You may only use values that are actually given. If the middle value isn't an input in the table, the composition can't be evaluated from the table. Say so; don't guess.
“To find $f(g(1))$ on the graph, I look for where $g$ equals 1.”

That's running $g$ backwards. $f(g(1))$ starts at the input $x = 1$ on the graph of $g$ : go to $x = 1$ , read the height $g(1) = 3$ , then use 3 as the input of $f$ .

Lock it in

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10 cards · Composition and inverses

Start

Recap card

5 lines to re-read the night before.

01
$f(g(x))$ = ( $f$ ∘ $g)( x$ ): apply $g$ first, then $f$ . Work from the inside out.
02
Order matters: $f(g(x))$ and $g(f(x))$ are usually different. Composition is not multiplication.
03
From a table or a graph, find the inside value first, then use it as the input of the outside function.
04
If the middle value is not in the domain of the outside function (or not given), the composition is undefined or can't be found.
05
The pieces can come from any mix of formulas, tables, and graphs. Next, in 2.7B: composing formulas to get a new formula, the domain of a composition, and breaking a function apart into simpler pieces.

Practice 33 problems Next: Chapter 2.7B Composition of Functions

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Composition of Functions

Output In, Output Out

From Formulas

From a Table

From a Graph

Mixing Formulas, Tables, and Graphs

Practice

Recap card

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