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 Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.4 Inverse FunctionsWatch the full walkthrough 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 $y$ .

1Let

y = 2 x + 3

and make

x

the subject:

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

2So

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

Answer

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

Only one-to-one functions

An inverse exists only if $f$ is $one-to-one$ . 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 $domain and range swap$ : the domain of $f^{- 1}$ is the range of $f$ , and vice versa. Also, $f f^{- 1} (x) = x$ .

f^{- 1} is f reflected in y = x, f f^{- 1} (x) = x

Domain of

f^{- 1}

= range of

f

Formula recap

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

Finding the inverse.

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

Inverse undoes the function.

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.

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