Inverse and Composite Functions

What you'll learn

Use function notation such as $f (3)$ and $g (- 2)$ .
Solve equations where a function output is given.
Find and use inverse functions, written $f^{-1}(x)$ .
Work with composite functions such as $f g (x)$ and $g f (x)$ .

1. Function notation: inputs and outputs

Definition

Function notation

A function is a rule that takes an input and gives exactly one output. In $f (x)$ , the letter $f$ names the function, and $x$ is the input.

Think of $f (x)$ as “the output from function $f$ when the input is $x$ ”. To evaluate a function, substitute the given input wherever you see $x$ .

Example

Evaluating a function

For $h(x)=2x^2-6$ , find $h (4)$ and $h (- 4)$ .

Function-machine view of substituting 4 and -4 into the same rule, showing both inputs give the same output.

To find $h (4)$ , replace every $x$ with 4.
Calculate:
$h(4)=2(4)^2-6=32-6=26$
To find $h (- 4)$ , replace every $x$ with $- 4$ . Keep brackets around the negative input.
Calculate:
$h(-4)=2(-4)^2-6=32-6=26$
The outputs match here because $4)^2=4^2$ .

Common Mistake

Forgetting brackets with negatives

Use $4)^2$ , not $4^2$ . The first gives 16, but the second means “negative of $4^2$ ”.

2. Solving with function notation

When a question says solve $q (x) = 23$ , it means “find the input values that make the output 23”. Replace $q (x)$ with the formula for the function, then solve the equation.

Key Idea

Function notation is just shorthand

A statement like $f (x) = g (x)$ means “set the formula for $f$ equal to the formula for $g$ ”.

Example

Solving a function equation

Let $q(x)=3x^2-4$ . Solve $q (x) = 23$ .

The output 23 is traced backwards through the function machine to find the possible inputs.

Replace $q (x)$ with its formula:
$3x^2-4=23$
Rearrange until $x^2$ is on its own:
$3x2−4=233x2=27x2=9\begin{aligned} 3x^2-4&=23\\ 3x^2&=27\\ x^2&=9 \end{aligned}$
Square root both sides:
$x=±3x=\pm 3$

Common Mistake

Missing the negative solution

If $x^2=9$ , then $x = 3$ or $x = - 3$ . A squared input can hide two possible answers.

3. Inverse functions: undoing a rule

Definition

Inverse function

The inverse function $f^{-1}$ undoes the function $f$ . If $f (a) = b$ , then $f^{-1}(b)=a$ .

To find an inverse, use this routine: write $y =$ the function, swap $x$ and $y$ , then rearrange to make $y$ the subject.

Example

Finding and using an inverse

For $r (x) = 4 x - 3$ , find $r^{-1}(x)$ . Then solve $r^{-1}(x)=5$ .

The inverse of r reverses the operations of multiplying by 4 and subtracting 3.

Write the function using $y$ :
$y = 4 x - 3$
Swap $x$ and $y$ :
$x = 4 y - 3$
Rearrange to make $y$ the subject:
$x=4y−3x+3=4yy=x+34\begin{aligned} x&=4y-3\\ x+3&=4y\\ y&=\frac{x+3}{4} \end{aligned}$
So the inverse is:
$r−1(x)=x+34r^{-1}(x)=\frac{x+3}{4}$
Now set the inverse equal to 5:
$x+34=5\frac{x+3}{4}=5$
Solve:
$x+3=20x=17\begin{aligned} x+3&=20\\ x&=17 \end{aligned}$

Common Mistake

Inverse is not reciprocal

The symbol $f^{-1}(x)$ means “undo $f$ ”. It does not mean $1f(x)\frac{1}{f(x)}$ .

Common Mistake

Quadratic inverses

A quadratic like $x^2-9$ gives the same output for inputs 3 and -3, so its inverse needs a restricted domain. The domain is the set of allowed inputs. If no restriction is stated, GCSE questions often use the non-negative square root, such as $f−1(x)=x+9f^{-1}(x)=\sqrt{x+9}$ .

A quadratic can fail to have a simple inverse because two different inputs can lead to the same output.

4. Composite functions: functions in a chain

Definition

Composite function

A composite function applies one function and then another. The notation $g f (x)$ means $g (f (x))$ , so you do $f$ first and then $g$ .

Common Mistake

Composition is not multiplication

In $g f (x)$ , the letters are not multiplied together. The function nearest $x$ is done first.

Example

Finding composites and solving them

Let $f (x) = 2 x + 1$ and $g(x)=x^2$ . Find $f g (x)$ , $g f (x)$ , and solve $f g (x) = g f (x)$ .

The two composite chains use the same functions in opposite orders, so they usually give different expressions.

For $f g (x)$ , do $g$ first, then put the result into $f$ :
$fg(x)=f(x^2)=2x^2+1$
For $g f (x)$ , do $f$ first, then put the result into $g$ :
$gf(x)=g(2x+1)=(2x+1)^2$
Set the two composite functions equal:
$2x^2+1=(2x+1)^2$
Expand and rearrange:
$2x2+1=4x2+4x+10=2x2+4x0=2x(x+2)\begin{aligned} 2x^2+1&=4x^2+4x+1\\ 0&=2x^2+4x\\ 0&=2x(x+2) \end{aligned}$
Solve from the factors:
$\text{ or } x=-2$

5. Substituting expressions and “hence” solving

An expression is a piece of algebra, such as $x - 3$ . If a question asks for $f (x - 3)$ , substitute the whole expression wherever $x$ appears.

Tip

Use brackets first

When substituting an expression, put it in brackets before expanding. This prevents sign and squaring mistakes.

Example

Finding $f (x - 3)$ and solving

Let $f(x)=x^2+2$ . Find $f (x - 3)$ , then solve $f (x - 3) = 11$ .

Substituting the whole expression x-3 into f means the entire expression is squared before adding 2.

Substitute $x - 3$ into the rule:
$f(x-3)=(x-3)^2+2$
Use the equation $f (x - 3) = 11$ :
$x-3)^2+2=11$
Rearrange:
$x-3)^2=9$
Square root both sides:
$\text{ or } x-3=-3$
Solve each equation:
$\text{ or } x=0$

6. Constants inside function questions

A constant is a value that does not change. In function questions, a constant may be shown as a letter such as $k$ .

Example

Finding an unknown constant

Let $a (x) = 3 x - 4$ and $b(x)=kx^2$ . Given that $ab (2) = 20$ , find $k$ .

For ab(2), the inside function b is applied first, then its result is used as the input to a.

Start with the inside function because $ab (2) = a (b (2))$ .
Work out $b (2)$ :
$b(2)=k(2)^2=4k$
Put $4 k$ into function $a$ :
$a (4 k) = 3 (4 k) - 4 = 12 k - 4$
Use the given output:
$12 k - 4 = 20$
Solve for $k$ :
$12k=24k=2\begin{aligned} 12k&=24\\ k&=2 \end{aligned}$

Exam technique

In the exam

Read composite functions from right to left: in $g f (3)$ , do $f$ first.
Use brackets whenever you substitute a negative number or an expression like $x - 2$ .
When square rooting, consider both positive and negative answers unless a domain or context rules one out.

Self review

Check yourself

If $f (x) = 4 x - 7$ , what operation would $f^{-1}$ undo first?
Why are $f g (x)$ and $g f (x)$ usually different?
When solving $x-2)^2=25$ , why are there two possible starting equations?

Inverse and Composite Functions

What you'll learn

Use function notation such as $f (3)$ and $g (- 2)$ .
Solve equations where a function output is given.
Find and use inverse functions, written $f^{-1}(x)$ .
Work with composite functions such as $f g (x)$ and $g f (x)$ .

1. Function notation: inputs and outputs

Definition

Function notation

A function is a rule that takes an input and gives exactly one output. In $f (x)$ , the letter $f$ names the function, and $x$ is the input.

Think of $f (x)$ as “the output from function $f$ when the input is $x$ ”. To evaluate a function, substitute the given input wherever you see $x$ .

Example

Evaluating a function

For $h(x)=2x^2-6$ , find $h (4)$ and $h (- 4)$ .

Function-machine view of substituting 4 and -4 into the same rule, showing both inputs give the same output.

To find $h (4)$ , replace every $x$ with 4.
Calculate:
$h(4)=2(4)^2-6=32-6=26$
To find $h (- 4)$ , replace every $x$ with $- 4$ . Keep brackets around the negative input.
Calculate:
$h(-4)=2(-4)^2-6=32-6=26$
The outputs match here because $4)^2=4^2$ .

Common Mistake

Forgetting brackets with negatives

Use $4)^2$ , not $4^2$ . The first gives 16, but the second means “negative of $4^2$ ”.

2. Solving with function notation

When a question says solve $q (x) = 23$ , it means “find the input values that make the output 23”. Replace $q (x)$ with the formula for the function, then solve the equation.

Key Idea

Function notation is just shorthand

A statement like $f (x) = g (x)$ means “set the formula for $f$ equal to the formula for $g$ ”.

Example

Solving a function equation

Let $q(x)=3x^2-4$ . Solve $q (x) = 23$ .

The output 23 is traced backwards through the function machine to find the possible inputs.

Replace $q (x)$ with its formula:
$3x^2-4=23$
Rearrange until $x^2$ is on its own:
$3x2−4=233x2=27x2=9\begin{aligned} 3x^2-4&=23\\ 3x^2&=27\\ x^2&=9 \end{aligned}$
Square root both sides:
$x=±3x=\pm 3$

Common Mistake

Missing the negative solution

If $x^2=9$ , then $x = 3$ or $x = - 3$ . A squared input can hide two possible answers.

3. Inverse functions: undoing a rule

Definition

Inverse function

The inverse function $f^{-1}$ undoes the function $f$ . If $f (a) = b$ , then $f^{-1}(b)=a$ .

To find an inverse, use this routine: write $y =$ the function, swap $x$ and $y$ , then rearrange to make $y$ the subject.

Example

Finding and using an inverse

For $r (x) = 4 x - 3$ , find $r^{-1}(x)$ . Then solve $r^{-1}(x)=5$ .

The inverse of r reverses the operations of multiplying by 4 and subtracting 3.

Write the function using $y$ :
$y = 4 x - 3$
Swap $x$ and $y$ :
$x = 4 y - 3$
Rearrange to make $y$ the subject:
$x=4y−3x+3=4yy=x+34\begin{aligned} x&=4y-3\\ x+3&=4y\\ y&=\frac{x+3}{4} \end{aligned}$
So the inverse is:
$r−1(x)=x+34r^{-1}(x)=\frac{x+3}{4}$
Now set the inverse equal to 5:
$x+34=5\frac{x+3}{4}=5$
Solve:
$x+3=20x=17\begin{aligned} x+3&=20\\ x&=17 \end{aligned}$

Common Mistake

Inverse is not reciprocal

The symbol $f^{-1}(x)$ means “undo $f$ ”. It does not mean $1f(x)\frac{1}{f(x)}$ .

Common Mistake

Quadratic inverses

A quadratic can fail to have a simple inverse because two different inputs can lead to the same output.

4. Composite functions: functions in a chain

Definition

Composite function

A composite function applies one function and then another. The notation $g f (x)$ means $g (f (x))$ , so you do $f$ first and then $g$ .

Common Mistake

Composition is not multiplication

In $g f (x)$ , the letters are not multiplied together. The function nearest $x$ is done first.

Example

Finding composites and solving them

Let $f (x) = 2 x + 1$ and $g(x)=x^2$ . Find $f g (x)$ , $g f (x)$ , and solve $f g (x) = g f (x)$ .

The two composite chains use the same functions in opposite orders, so they usually give different expressions.

For $f g (x)$ , do $g$ first, then put the result into $f$ :
$fg(x)=f(x^2)=2x^2+1$
For $g f (x)$ , do $f$ first, then put the result into $g$ :
$gf(x)=g(2x+1)=(2x+1)^2$
Set the two composite functions equal:
$2x^2+1=(2x+1)^2$
Expand and rearrange:
$2x2+1=4x2+4x+10=2x2+4x0=2x(x+2)\begin{aligned} 2x^2+1&=4x^2+4x+1\\ 0&=2x^2+4x\\ 0&=2x(x+2) \end{aligned}$
Solve from the factors:
$\text{ or } x=-2$

5. Substituting expressions and “hence” solving

An expression is a piece of algebra, such as $x - 3$ . If a question asks for $f (x - 3)$ , substitute the whole expression wherever $x$ appears.

Tip

Use brackets first

When substituting an expression, put it in brackets before expanding. This prevents sign and squaring mistakes.

Example

Finding $f (x - 3)$ and solving

Let $f(x)=x^2+2$ . Find $f (x - 3)$ , then solve $f (x - 3) = 11$ .

Substituting the whole expression x-3 into f means the entire expression is squared before adding 2.

Substitute $x - 3$ into the rule:
$f(x-3)=(x-3)^2+2$
Use the equation $f (x - 3) = 11$ :
$x-3)^2+2=11$
Rearrange:
$x-3)^2=9$
Square root both sides:
$\text{ or } x-3=-3$
Solve each equation:
$\text{ or } x=0$

6. Constants inside function questions

A constant is a value that does not change. In function questions, a constant may be shown as a letter such as $k$ .

Example

Finding an unknown constant

Let $a (x) = 3 x - 4$ and $b(x)=kx^2$ . Given that $ab (2) = 20$ , find $k$ .

For ab(2), the inside function b is applied first, then its result is used as the input to a.

Start with the inside function because $ab (2) = a (b (2))$ .
Work out $b (2)$ :
$b(2)=k(2)^2=4k$
Put $4 k$ into function $a$ :
$a (4 k) = 3 (4 k) - 4 = 12 k - 4$
Use the given output:
$12 k - 4 = 20$
Solve for $k$ :
$12k=24k=2\begin{aligned} 12k&=24\\ k&=2 \end{aligned}$

Exam technique

In the exam

Read composite functions from right to left: in $g f (3)$ , do $f$ first.
Use brackets whenever you substitute a negative number or an expression like $x - 2$ .
When square rooting, consider both positive and negative answers unless a domain or context rules one out.

Self review

Check yourself

If $f (x) = 4 x - 7$ , what operation would $f^{-1}$ undo first?
Why are $f g (x)$ and $g f (x)$ usually different?
When solving $x-2)^2=25$ , why are there two possible starting equations?

Inverse and Composite Functions

What you'll learn

1. Function notation: inputs and outputs

2. Solving with function notation

3. Inverse functions: undoing a rule

4. Composite functions: functions in a chain

5. Substituting expressions and “hence” solving

6. Constants inside function questions

Recap questions

You've reached the end

Inverse and Composite Functions

What you'll learn

1. Function notation: inputs and outputs

2. Solving with function notation

3. Inverse functions: undoing a rule

4. Composite functions: functions in a chain

5. Substituting expressions and “hence” solving

6. Constants inside function questions

Recap questions

You've reached the end

Inverse and Composite Functions

What you'll learn

1. Function notation: inputs and outputs

2. Solving with function notation

3. Inverse functions: undoing a rule

4. Composite functions: functions in a chain

5. Substituting expressions and “hence” solving

6. Constants inside function questions

Recap questions

You've reached the end

Inverse and Composite Functions Revision Guide

Inverse and Composite Functions

What you'll learn

1. Function notation: inputs and outputs

2. Solving with function notation

3. Inverse functions: undoing a rule

4. Composite functions: functions in a chain

5. Substituting expressions and “hence” solving

6. Constants inside function questions

Recap questions

You've reached the end

Inverse and Composite Functions Revision Guide