2.4PureStretch

Inverse Functions

The inverse function $f^{-1}$ undoes $f$ : feed an output back in and you get the original input. Only one-to-one functions have inverses, and the graph of $f^{-1}$ is the reflection of $f$ in the line $y = x$ .

30 min Functions and Graphs

Edexcel A level Maths: 2.4 Inverse FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Find the inverse of a function algebraically
Know that only one-to-one functions are invertible
Swap domain and range for the inverse
Relate f and f⁻¹ graphically

Finding the inverse

To find $f^{-1}(x)$ : write $y = f(x)$ , swap $x$ and $y$ (or rearrange to make $x$ the subject), then solve for the new $y$ .

Example 1Find the inverse of

f(x) = 2x + 3

1Let

y = 2x + 3

and make

x

the subject:

x = \dfrac{y - 3}{2}

2So

f^{-1}(x) = \dfrac{x - 3}{2}

Answer

f^{-1}(x) = \dfrac{x - 3}{2}

Only one-to-one functions

An inverse exists only if $f$ is $\textbf{one-to-one}$ . A many-to-one function (like $x^2$ ) must have its domain restricted first, otherwise the “inverse” would not be a function.

Tip — If a function fails the horizontal-line test, restrict its domain before inverting.

Graphs and domain/range

The graph of $f^{-1}$ is the reflection of $f$ in the line $y = x$ . The $\textbf{domain and range swap}$ : the domain of $f^{-1}$ is the range of $f$ , and vice versa. Also, $ff^{-1}(x) = x$ .

f^{-1} \text{ is } f \text{ reflected in } y = x, \quad ff^{-1}(x) = x

Domain of

f^{-1}

= range of

f

Formula recap

y = f(x) \Rightarrow \text{swap, solve for } f^{-1}

Finding the inverse.

ff^{-1}(x) = f^{-1}f(x) = x

Inverse undoes the function.

\text{reflect in } y = x

Graph of the inverse.

Common mistakes to avoid

Inverting a many-to-one function without restricting the domain.

Only one-to-one functions are invertible — restrict the domain first.

Keeping the same domain for f⁻¹.

The domain of f⁻¹ is the range of f (they swap).

Key takeaways

f⁻¹ undoes f: ff⁻¹(x) = x.
Find it by making x the subject (or swapping x and y).
Only one-to-one functions have inverses; the graph reflects in y = x and domain/range swap.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The inverse of $f(x) = 2x + 3$ is:
- $\dfrac{x-3}{2}$
- $\dfrac{x+3}{2}$
- $2x - 3$
- $\dfrac{1}{2x+3}$
Show answer and explanation
Answer: $\dfrac{x-3}{2}$
Make x the subject.
A function has an inverse only if it is:
- One-to-one
- Many-to-one
- Quadratic
- Increasing
Show answer and explanation
Answer: One-to-one
Otherwise the inverse is not a function.
The graph of $f^{-1}$ is $f$ reflected in:
- $y = x$
- the x-axis
- the y-axis
- the origin
Show answer and explanation
Answer: $y = x$
Reflection in y = x.
$ff^{-1}(x)$ equals:
- $x$
- $f(x)^{2}$
- $1$
- $0$
Show answer and explanation
Answer: $x$
The inverse undoes f.
The domain of $f^{-1}$ is the ___ of $f$ .
- Range
- Domain
- Gradient
- Intercept
Show answer and explanation
Answer: Range
Domain and range swap.
To invert $f(x) = x^{2}$ , you must first:
- Restrict the domain
- Square it again
- Add a constant
- Differentiate
Show answer and explanation
Answer: Restrict the domain
It is many-to-one otherwise.

Next step

Keep practising

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