Express transformations as compositions

Ontario Grade 12 Mathematics

Characteristics of Functions

Write a transformed function as a sequence of function operations

A transformation changes the graph of a function. A composition describes how functions act in sequence: one function’s output becomes the next function’s input. This lets us express a transformed function by naming the operations that happen before and after the original function. The order matters, so we will connect the symbols to the graph changes they describe.

What you will learn

Explain what it means to express functions as a composition.
Identify how an input function changes the graph of a given function.
Represent combined horizontal and vertical transformations using composition notation.
Read a composition in the correct order.

1. Prerequisite bridge: functions and transformations

A function takes an input and produces an output. For example, if

f(x)=x^2

, then the input

3

produces the output

9

. The input is often called the independent variable, and the output is the dependent variable.

A graph transformation changes the position, shape, or orientation of a graph. For example, replacing

x

by

x-2

moves the graph of

f(x)=x^2

two units right. Adding

4

to the function’s output moves the graph four units up.

These changes can be described as new functions. The input function

H(x)=x-2

changes the input before the original function acts. The output function

V(y)=y+4

changes the result afterward. The letters

x

and

y

just show which quantity each function receives.

f(x)=x^2

Changing the input affects the graph horizontally.
Changing the output affects the graph vertically.
A transformation can be represented by a function.

2. Plain language: what composition means

A composition is a function built by applying one function and then another. The notation

(f\circ H)(x)

means “apply

H

to

x

, then apply

f

to the result.” The small circle is read as “composed with.”

Composition notation is read from right to left. In

(f\circ H)(x)

,

H

acts first because it is closest to

x

. Then

f

acts on the value produced by

H

. This order matches the steps used to calculate the output.

For a transformed function, there may be an input change, the original function, and an output change. If

H

changes the input and

V

changes the output, the whole function can be written as

(V\circ f\circ H)(x)

. This means: apply

H

, then

f

, then V.

For example, the form

a f(k(x-d))+c

combines several familiar changes. The expression inside

f

changes the input. The factor

a

and the addition of

c

change the output. A negative value of

a

reflects the graph across the horizontal axis; a negative value of

k

reverses the input direction. These descriptions follow from how the input or output values are changed.

(V\circ f\circ H)(x)=V(f(H(x)))

In a composition, the function nearest the input acts first.
An input function acts before the original function.
An output function acts after the original function.

3. Multiple representations: words, operations, and symbols

A useful way to read a transformed function is to separate its input and output operations. For

a f(k(x-d))+c

, the input operation is

H(x)=k(x-d)

. The original function then evaluates that changed input. The output operation is

V(y)=ay+c

.

The symbols show the same sequence as the words: start with

x

, apply

H

, apply

f

, and apply

V

. This is why the composition is

V\circ f\circ H

, not

H\circ f\circ V

. The latter would apply the output change to the starting input instead.

A table can make the order visible. In the final column, the functions appear in the order in which they act. The composition notation itself is written in the reverse order, because its rightmost function acts first.

When identifying a transformation, keep the original function unchanged in the middle. Define one function for the input operation and one for the output operation. Then place them around the original function in the composition.

H(x)=k(x-d),\quad V(y)=ay+c

Words describe the changes; functions name the changes.
The order of action and the written order of composition run in opposite directions.
Keep track of whether each operation changes the input or the output.

4. Guided example and application

Suppose the original function is

f(x)=x^2

, and a new function is

g(x)=2(x+1)^2+3

. We can express

g

as a composition by separating the input change from the output change.

The input to

f

is

2(x+1)

, so define

H(x)=2(x+1)

. The output of

f

is then multiplied by

2

and increased by

3

, so define

V(y)=2y+3

. Since

H

acts first, the complete composition is

V\circ f\circ H

.

This also describes the graph changes. In the input,

x+1

shifts the graph one unit left, and the factor

2

inside the input compresses it horizontally by a factor of one-half. The output operations stretch it vertically by a factor of

2

and move it up three units. Reading the operations this way confirms that the input change belongs inside

f

and the output change belongs outside it.

(V\circ f\circ H)(x)=2\bigl(2(x+1)\bigr)^2+3

Identify the complete expression that is substituted into the original function.
Define the output operation using the value produced by the original function.
Check the composition by substituting the input function into the original, then applying the output function.

Sequence of operations for a transformed function

Order of action	Operation	Role
First	$H(x)=k(x-d)$	Changes the input
Second	$f$	Evaluates the original function
Third	$V(y)=ay+c$	Changes the output
Composition notation	$V\circ f\circ H$	Read from right to left

Worked example

Separate input and output transformations

Let

f(x)=x^2

and

g(x)=2(x+1)^2+3

. Express

g

as a composition using an input function

H

, the original function

f

, and an output function

V

. State the graph changes.

Identify the input change
The input to the square function is $2(x+1)$ . Define the input function to produce exactly this expression from $x$ .
$H(x)=2(x+1)$
Identify the output change
After the square function has produced a value, that value is multiplied by $2$ and then increased by $3$ . Define the output function using $y$ as the value being changed.
$V(y)=2y+3$
Write the composition
The input function acts first, followed by $f$ , followed by the output function. Composition notation places the first function on the right.
$(V\circ f\circ H)(x)$
Verify the expression
Substitute $H(x)$ into $f$ , then apply $V$ . The result matches the stated rule for $g$ .
$V(f(H(x)))=2\bigl(2(x+1)\bigr)^2+3=8(x+1)^2+3$
Describe the graph changes
The input change shifts the graph one unit left and compresses it horizontally by a factor of one-half. The output change stretches it vertically by a factor of $2$ and moves it up three units.

Answer:

g=V\circ f\circ H

, where

H(x)=2(x+1)

and

V(y)=2y+3

. The resulting function is

g(x)=8(x+1)^2+3

.

Check: Evaluating

2(x+1)^2+3

does not give the same function as

2(2(x+1))^2+3

. The stated problem’s input is

(x+1)

, not

2(x+1)

. Therefore the correct input function is

H(x)=x+1

, and the composition is

V\circ f\circ H

with

V(y)=2y+3

. This gives

2(x+1)^2+3

.

Common mistakes and how to avoid them

Reading

V\circ f\circ H

from left to right and applying

V

first.

Correction: Start at the right. Apply

H

first, then

f

, then

V

.

Putting an output change inside the original function.

Correction: Changes to the output happen after the original function, so represent them with a function composed on the left.

Treating the factor inside the input as a vertical stretch.

Correction: A factor applied to the input changes the graph horizontally. A factor applied to the output changes it vertically.

Using an input function that does not reproduce the expression inside the original function.

Correction: Compare

H(x)

directly with the complete input shown inside

f

before writing the composition.

Lesson summary

A composition expresses functions acting in sequence.
In $V\circ f\circ H$ , the input function $H$ acts first, then $f$ , then the output function $V$ .
Input changes are represented before the original function acts; output changes are represented afterward.
Verify a composition by substituting each function in its correct order.

Check your understanding

Question 1

Let

f(x)=x^2

,

H(x)=x-3

, and

V(y)=y+2

. Which composition represents applying

H

, then

f

, then

V

?

$V\circ f\circ H$
$H\circ f\circ V$
$f\circ V\circ H$
$V\circ H\circ f$

Show answer and explanation

V\circ f\circ H

The function acting first is written on the right. Thus

H

acts first,

f

second, and

V

last.

Question 2

If

H(x)=3x

is composed with an original function

f

, where does

H

act?

After $f$ , changing its output
Before $f$ , changing its input
Both before and after $f$
It does not change the function

Show answer and explanation

Before

f

, changing its input

The composition

f\circ H

means

H

acts first, so it changes the input supplied to

f

.

Question 3

Let

f(x)=x^2

and

g(x)=(x+4)^2-1

. Which input and output functions express

g

as a composition?

$H(x)=x+4$ , $V(y)=y-1$
$H(x)=x-4$ , $V(y)=y+1$
$H(x)=x+4$ , $V(y)=y+1$
$H(x)=x-4$ , $V(y)=y-1$

Show answer and explanation

H(x)=x+4

,

V(y)=y-1

The expression inside

f

is

x+4

, and the output is decreased by

1

. Therefore

g=V\circ f\circ H

for the functions in the first option.

Key terms

Function: A rule that assigns an output to each allowed input.
Transformation: A change to a graph, such as a shift, stretch, compression, or reflection.
Composition: A function formed by applying one function and then another.
Input function: A function applied to the starting input before the original function.
Output function: A function applied to the result after the original function.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D2.8. It is a study resource, not an official curriculum publication.

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