Compose functions numerically and graphically

Ontario Grade 12 Mathematics

Characteristics of Functions

Follow one function’s output into another function

A function takes an input and gives an output. When two functions are composed, the output from the first function becomes the input to the second. You can show this process with numbers, tables, graphs, or symbols. The order matters: changing which function acts first can change the result.

What you will learn

Explain what it means to compose two functions.
Find values of a composite function using a numerical rule or a table.
Read a composite function’s values from information shown on graphs.
Write and interpret composite-function notation.

1. Prerequisite bridge: inputs and outputs

A function is a rule that assigns an output to each allowed input. The input is often called

x

. The output of a function named

f

is written

f(x)

. For example, if

f(x)=2x+1

, then

f(3)=7

because replacing

x

with

3

gives

2(3)+1=7

.

A function can also be described by a table or a graph. In a table, match an input with its output. On a graph, the point

(a,b)

shows that the output at input

a

is

b

. In other words, that point tells you

f(a)=b

.

These ideas are all ways to represent the same input-output relationship. To compose functions, you need to be able to find an output from the first function and use it as an input to the next.

f(a)=b

The input is the value placed into a function.
The output is the value produced by the function.
A graph point $(a,b)$ represents the input-output pair $f(a)=b$ .

2. Plain language and numerical composition

To compose two functions, apply one function and then apply the other to the result. The inside function acts first. For example, in

f(g(x))

, start with

x

, find

g(x)

, and use that output as the input to

f

.

The notation

(f\circ g)(x)

is read as “

f

composed with

g

at

x

.” It means the same thing as

f(g(x))

. The symbol

\circ

names the composition; it does not mean multiplication.

Suppose

g(x)=x+2

and

f(x)=3x

. At input

4

, the first function gives

g(4)=6

. That output becomes the input to

f

, so the final output is

f(6)=18

. Thus,

(f\circ g)(4)=18

.

The order can change the answer. If

f

acts first instead,

f(4)=12

, and then

g(12)=14

. So

(g\circ f)(4)=14

, not

18

. Always follow the order shown by the notation.

A numerical table can show the same process. For each starting input, read the output from

g

, then use that output to look up an output from

f

. A composition only has a value when the output of the first function is an allowed input for the second.

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

In $f(g(x))$ , apply $g$ first and $f$ second.
The output of the first function becomes the next input.
Reversing the order may produce different values.

3. Reading composition from graphs

A graph can provide the values needed for a composition even when no equation is given. To find

(f\circ g)(a)

, first locate input

a

on the graph of

g

and read its output. Then use that output as the input on the graph of

f

.

For example, if the graph of

g

shows the point

(2,5)

, then

g(2)=5

. If the graph of

f

shows the point

(5,3)

, then

f(5)=3

. Together, these readings give

(f\circ g)(2)=3

.

A useful way to keep the readings organized is to follow the chain: starting input, output of

g

, output of

f

. When reading a graph, use the horizontal coordinate as the input and the vertical coordinate as the output. Do not use the output of

g

as the final answer; it is the input for

f

.

A graph or a list of points may show only selected values. You can find a composite value only when the needed outputs can be read from the given information. Do not assume a value between plotted points unless the graph or question gives enough information to read it.

(f\circ g)(a)=f(g(a))

Read the graph of the inside function first.
Use its output as the input on the other function’s graph.
Each point $(a,b)$ on a function graph means that the function’s value at $a$ is $b$ .

4. Application and careful checking

Composition describes a process with two stages. Imagine a machine that changes an input according to

g

, followed by a second machine that changes the new value according to

f

. The combined process is represented by

f(g(x))

. This interpretation works whether the rules are given as equations, tables, or graphs.

When solving a question, first identify which function acts first. Then find its output at the starting input. Check that this output is an allowed input for the next function. Finally, find the next output and state it as the value of the composite function.

You can check your reasoning by writing the input-output chain in words or in a small table. If the final answer is the output of the first function rather than the output of the second, the process stopped too soon.

(f\circ g)(a)=f\bigl(g(a)\bigr)

The function inside the parentheses acts first.
The first output must be usable as an input to the second function.
The final value is the output of the second function.

Graph readings used in the example

Starting input	Output from $g$	Output from $f$	Composite output
$-2$	$1$	$4$	$4$
$0$	$3$	$-1$	$-1$
$2$	$1$	$4$	$4$

Worked example

Compose functions from graph readings

The graph of

g

contains the points

(-2,1)

,

(0,3)

, and

(2,1)

. The graph of

f

contains the points

(1,4)

and

(3,-1)

. Find

(f\circ g)(-2)

,

(f\circ g)(0)

, and

(f\circ g)(2)

.

Read the first graph
Use the graph of $g$ to find the output for each starting input. The points show that $g(-2)=1$ , $g(0)=3$ , and $g(2)=1$ .
$g(-2)=1,\quad g(0)=3,\quad g(2)=1$
Use those outputs as inputs
Now read the graph of $f$ . The input $1$ gives output $4$ , and the input $3$ gives output $-1$ . These are the values needed because the outputs from $g$ were $1$ or $3$ .
$f(1)=4,\quad f(3)=-1$
Complete each composition
For each starting input, use its $g$ output as the input to $f$ . This gives the final values of the three compositions.
$(f\circ g)(-2)=4,\quad (f\circ g)(0)=-1,\quad (f\circ g)(2)=4$
Represent the composite values
Each result can be shown as a point on the graph of the composite function. The horizontal coordinate is the original input, and the vertical coordinate is the final output.
$(-2,4),\quad (0,-1),\quad (2,4)$

Answer:

(f\circ g)(-2)=4

,

(f\circ g)(0)=-1

, and

(f\circ g)(2)=4

. The corresponding points are

(-2,4)

,

(0,-1)

, and

(2,4)

.

Check: For the input

0

, the graph of

g

gives

3

, and the graph of

f

gives

-1

at input

3

. The final value is therefore

-1

, not

3

.

Common mistakes and how to avoid them

Applying the function named first in the notation first. For

f(g(x))

, this would mean using

f

before

g

.

Correction: Start with the inside function,

g

. Use its output as the input to

f

.

Treating the composition symbol as multiplication.

Correction: The symbol

\circ

means that one function is applied after another. It does not mean multiply the function values.

Stopping at the output of the first function.

Correction: That output is only an intermediate value. Use it as the input to the second function to get the composite output.

Assuming reversing the functions leaves the result unchanged.

Correction: Check the order shown. In general, applying

g

and then

f

can give a different result from applying

f

and then

g

.

Lesson summary

Composition connects the output of one function to the input of another.
In $f(g(x))$ , apply $g$ first and $f$ second.
Use tables or graphs by reading the first function’s output, then using it as the next input.
The final answer is the output after both functions have acted.

Check your understanding

Question 1

Suppose

g(2)=5

and

f(5)=9

. What is

(f\circ g)(2)

?

$2$
$5$
$9$
It cannot be determined from the information given.

Show answer and explanation

9

First,

g

sends

2

to

5

. Then

f

sends

5

to

9

, so the composite value is

9

.

Question 2

If

f(4)=7

and

g(7)=1

, what is

(g\circ f)(4)

?

$1$
$4$
$7$
It cannot be determined from the information given.

Show answer and explanation

1

In

g(f(4))

, apply

f

first to get

7

, then apply

g

to

7

to get

1

.

Key terms

Function: A rule that assigns an output to each allowed input.
Composition: A process in which the output of one function is used as the input to another.
Composite function: A function made by applying one function and then another, such as $(f\circ g)(x)$ .
Input-output pair: A pair of values showing an input and its matching output. On a graph, it is shown as a point.

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