Year 12 Functions and relations: Composite functions and equations

Composite functions are built by feeding the output of one function into another. This is one of the most important ideas in the Year 12 Functions course because it shows how function rules can be layered. A composition question may ask you to calculate a new formula, describe the effect graphically, or solve an equation involving the composition. The skill is mostly about reading the order correctly. The inner function acts first, and the outer function acts second.

Work through the sections in order, then try the linked Year 12 skill quizzes after each section.

f (g (x))

, the function

g

happens first because its output becomes the input of

f

Main method

Identify the inner function clearly.
Substitute that inner function into the outer one.
Simplify only after the substitution is correct.
If solving an equation, watch domain restrictions and possible invalid values.

Skill quiz: Composite functions

Practise the skills used in this section before continuing.

Composite functions

Worked example 1

Problem: Let

f (x) = x^{2} + 1

and

g (x) = 3 x - 2

. Find

f (g (x))

The inner function is $g (x) = 3 x - 2$ .
Substitute into $f$ : $f (g (x)) = {(3 x - 2)}^{2} + 1$ .
Expand if needed: $9 x^{2} - 12 x + 4 + 1$ .
Simplify: $f (g (x)) = 9 x^{2} - 12 x + 5$ .

Answer:

f (g (x)) = 9 x^{2} - 12 x + 5

Skill quiz: Composite functions

Try these questions using the method from the worked example.

Composite functions

Worked example 2

Problem: Let

f (x) = x + 4

and

g (x) = x^{2}

. Solve

f (g (x)) = 13

First form the composition: $f (g (x)) = x^{2} + 4$ .
Set this equal to 13: $x^{2} + 4 = 13$ .
Rearrange: $x^{2} = 9$ .
So $x = 3$ or $x = - 3$ .

Answer: the solutions are

x = 3

and

x = - 3

Skill quiz: Composite function equations

Try these questions using the method from the worked example.

Level 1: Composite function equations

How equations fit in

Composite-function equations are really a natural extension of basic composition. Once you can write the composition correctly, the solving stage usually becomes standard algebra. The challenge is that a small substitution error at the start makes every later line wrong. That is why writing the composition carefully matters more than rushing into the equation solving.

Skill quiz: Composite function equations

Practise the skills used in this section before continuing.

Level 2: Composite function equations

Worked example 3

Problem: Let

f (x) = 2 x + 1

and

g (x) = x - 5

. Compare

f (g (x))

and

g (f (x))

First, $f (g (x)) = 2 (x - 5) + 1 = 2 x - 9$ .
Next, $g (f (x)) = (2 x + 1) - 5 = 2 x - 4$ .
The two rules are not equal.
This shows that composition usually depends on order.

Answer:

f (g (x)) = 2 x - 9

but

g (f (x)) = 2 x - 4

, so they are different.

Skill quiz: Composite functions

Try these questions using the method from the worked example.

Composite functions

Common traps

Reading $f (g (x))$ and $g (f (x))$ as if they must be the same.
Treating the composition like multiplication.
Solving the resulting equation without checking restrictions from the original functions.

Skill quiz: Domain and range of composite functions

Use these questions to check the errors described above.

Domain and range of composite functions

Revision focus

A strong composition habit is to mark the inner function before you substitute. Once that is clear, the order of action becomes much easier to keep straight. This is especially important in composite-function equations, where the algebra feels familiar and can tempt you to skip the setup.

It also helps to think about composition graphically. The inner function changes the input first, so its effect often behaves like a horizontal change from the perspective of the outer graph. That connection between composition and transformation is one of the main ideas unifying the whole Functions topic.

Skill quizzes: Composite functions

Apply the checks from this section while working through the quiz.