Edexcel GCSE Mathematics · 1MA1

Composite Functions

What you need to know

Edexcel 1MA1 · A7 Functions Check the specification (PDF) (opens in a new tab)

Interpret simple expressions as functions with inputs and outputs.
Interpret the reverse process as the 'inverse function'.
Interpret the succession of two functions as a 'composite function'.
Use formal function notation.

Joining two function processes

A function takes an input, applies a rule and produces an output. For example, if $g(x)=x^2$ , then $g$ takes a number and squares it. If $f(x)=2x+3$ , then $f$ doubles a number and adds 3.

A composite function is made by applying these functions one after the other. The output of the first function becomes the input of the second. This succession of two functions can be written

$fg(x)=f(g(x)).$

Read this as “ $f$ of $g$ of $x$ ”. The function closest to $x$ acts first, so $g$ is applied before $f$ .

Two vertically stacked function-machine chains compare composition order. In the top chain, input x passes through g first, so the intermediate value g(x) becomes the input to f, producing f(g(x)) = fg(x). In the bottom chain, the same input passes through f first, so f(x) becomes the input to g, producing g(f(x)) = gf(x). The final comparison states that these outputs are generally unequal. — A composite is read from the function closest to the input: $fg(x)=f(g(x))$ applies $g$ first, whereas $gf(x)=g(f(x))$ applies $f$ first.

Following an input through both functions

Using $f(x)=2x+3$ and $g(x)=x^2$ , begin with an input of $-1$ .

First apply $g$ :

$g(-1)=(-1)^2=1.$

The output 1 now becomes the input of $f$ :

$f(1)=2(1)+3=5.$

Therefore

$fg(-1)=f(g(-1))=5.$

The nested brackets record the order of the two processes: work from the inside out.

Forming an algebraic composite

A composite can also be found as an expression in $x$ . For the same functions,

$fg(x)=f(g(x)).$

The input to $f$ is now $g(x)=x^2$ . Replace every occurrence of the input in the rule for $f$ by $x^2$ :

$fg(x)=f(x^2)=2x^2+3.$

It can help to imagine the rule for $f$ written with an empty input box: “double the box, then add 3”. The whole expression $g(x)$ goes into that box.

For a rule containing several occurrences of its input, brackets are especially important. Suppose

$f(x)=x^2+3x-2 \quad\text{and}\quad g(x)=x+3.$

Then

$fg(x)=f(x+3)=(x+3)^2+3(x+3)-2.$

Expanding and collecting like terms gives

$fg(x)=x^2+9x+16.$

Every $x$ in the rule for $f$ has been replaced by the complete expression $x+3$ .

Why the order matters

Reversing the letters reverses the order of the functions:

$gf(x)=g(f(x)).$

Return to $f(x)=2x+3$ and $g(x)=x^2$ . Applying $f$ first gives

$gf(x)=g(2x+3)=(2x+3)^2.$

This is different from

$fg(x)=2x^2+3.$

For instance, $gf(-1)=1$ , whereas $fg(-1)=5$ . In general, $fg(x)$ and $gf(x)$ are not equal, although particular pairs of functions can sometimes produce the same result in both orders.

When a composite is permitted

The intermediate output must be an allowed input for the next function. In $fg(x)$ , the output from $g$ must be suitable as an input to $f$ . For ordinary linear and quadratic rules this often causes no difficulty, but restrictions matter for rules such as division, where a denominator cannot be zero. The two-stage process must make sense at every step.

Meaning and order

A composite applies two functions successively:

$fg(x)=f(g(x)).$

Work from the inside out: apply $g$ first, then $f$ . Reversing the notation reverses the process:

$gf(x)=g(f(x)).$

Usually, $fg(x)\ne gf(x)$ .

Algebraic method

Identify the inner function.
Substitute its complete expression into every occurrence of the variable in the outer function.
Use brackets, then expand or simplify if required.

For a numerical input, pass the value through the inner function and use its output as the input of the outer function.

Validity check

For $fg(x)$ to be defined, the output of $g$ must be an allowed input of $f$ . Check restrictions such as division by zero.

Exam tips

1
Read a composite from right to left: in $fg(x)=f(g(x))$ , apply $g$ first.
2
When substituting one expression into another function, put the entire expression in brackets before expanding.
3
Do not assume $fg(x)=gf(x)$ ; calculate each composition in the stated order.
4
If a particular input is given, either follow the two function steps numerically or form the composite first and then substitute. Both methods should agree.

Ask the tutor about this note

It can see this note and what your specification asks for.

Answers are free with an account.

1 free note left this week

Get more free notes

How well do you know this now?

Your answer sets when this note comes back for review, or marks it complete.

Was this note helpful?

Last updated 5 Oct 2026

Key terms(4)

Function: A rule that maps each permitted input to one output.
Composite function: A function formed by applying one function and then applying another function to its output.
Input: The value put into a function.
Output: The value produced by a function.

Previous noteInverse Functions Next topic: Graphical MethodsCoordinates

Back to Functions

Put your knowledge into practice — try past paper questions for Mathematics

Edexcel GCSE Mathematics · 1MA1

Composite Functions

What you need to know

Edexcel 1MA1 · A7 Functions Check the specification (PDF) (opens in a new tab)

Interpret simple expressions as functions with inputs and outputs.
Interpret the reverse process as the 'inverse function'.
Interpret the succession of two functions as a 'composite function'.
Use formal function notation.

Joining two function processes

A function takes an input, applies a rule and produces an output. For example, if $g(x)=x^2$ , then $g$ takes a number and squares it. If $f(x)=2x+3$ , then $f$ doubles a number and adds 3.

A composite function is made by applying these functions one after the other. The output of the first function becomes the input of the second. This succession of two functions can be written

$fg(x)=f(g(x)).$

Read this as “ $f$ of $g$ of $x$ ”. The function closest to $x$ acts first, so $g$ is applied before $f$ .

Following an input through both functions

Using $f(x)=2x+3$ and $g(x)=x^2$ , begin with an input of $-1$ .

First apply $g$ :

$g(-1)=(-1)^2=1.$

The output 1 now becomes the input of $f$ :

$f(1)=2(1)+3=5.$

Therefore

$fg(-1)=f(g(-1))=5.$

The nested brackets record the order of the two processes: work from the inside out.

Forming an algebraic composite

A composite can also be found as an expression in $x$ . For the same functions,

$fg(x)=f(g(x)).$

The input to $f$ is now $g(x)=x^2$ . Replace every occurrence of the input in the rule for $f$ by $x^2$ :

$fg(x)=f(x^2)=2x^2+3.$

It can help to imagine the rule for $f$ written with an empty input box: “double the box, then add 3”. The whole expression $g(x)$ goes into that box.

For a rule containing several occurrences of its input, brackets are especially important. Suppose

$f(x)=x^2+3x-2 \quad\text{and}\quad g(x)=x+3.$

Then

$fg(x)=f(x+3)=(x+3)^2+3(x+3)-2.$

Expanding and collecting like terms gives

$fg(x)=x^2+9x+16.$

Every $x$ in the rule for $f$ has been replaced by the complete expression $x+3$ .

Why the order matters

Reversing the letters reverses the order of the functions:

$gf(x)=g(f(x)).$

Return to $f(x)=2x+3$ and $g(x)=x^2$ . Applying $f$ first gives

$gf(x)=g(2x+3)=(2x+3)^2.$

This is different from

$fg(x)=2x^2+3.$

For instance, $gf(-1)=1$ , whereas $fg(-1)=5$ . In general, $fg(x)$ and $gf(x)$ are not equal, although particular pairs of functions can sometimes produce the same result in both orders.

When a composite is permitted

Meaning and order

A composite applies two functions successively:

$fg(x)=f(g(x)).$

Work from the inside out: apply $g$ first, then $f$ . Reversing the notation reverses the process:

$gf(x)=g(f(x)).$

Usually, $fg(x)\ne gf(x)$ .

Algebraic method

Identify the inner function.
Substitute its complete expression into every occurrence of the variable in the outer function.
Use brackets, then expand or simplify if required.

For a numerical input, pass the value through the inner function and use its output as the input of the outer function.

Validity check

For $fg(x)$ to be defined, the output of $g$ must be an allowed input of $f$ . Check restrictions such as division by zero.

Exam tips

1
Read a composite from right to left: in $fg(x)=f(g(x))$ , apply $g$ first.
2
When substituting one expression into another function, put the entire expression in brackets before expanding.
3
Do not assume $fg(x)=gf(x)$ ; calculate each composition in the stated order.
4
If a particular input is given, either follow the two function steps numerically or form the composite first and then substitute. Both methods should agree.

Ask the tutor about this note

It can see this note and what your specification asks for.

Answers are free with an account.

1 free note left this week

Get more free notes

How well do you know this now?

Your answer sets when this note comes back for review, or marks it complete.

Was this note helpful?

Last updated 5 Oct 2026

Key terms(4)

Function: A rule that maps each permitted input to one output.
Composite function: A function formed by applying one function and then applying another function to its output.
Input: The value put into a function.
Output: The value produced by a function.

Previous noteInverse Functions Next topic: Graphical MethodsCoordinates

Back to Functions

Put your knowledge into practice — try past paper questions for Mathematics