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.

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.

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

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

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.

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.

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.

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.

Year 12 Functions and relations: Composite functions and equations

Main method

Worked example 1

Worked example 2

How equations fit in

Worked example 3

Common traps

Revision focus

Practice links