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How to Find an Inverse Function

An inverse undoes a function. To find it, swap the inputs and outputs.

The method

Replace $f (x)$ with $y$ .
Swap $x$ and $y$ .
Solve for $y$ .
Check that $f (f^{- 1} (x)) = x$ .

Worked examples

Example: $f (x) = 2 x + 3$

Swap: $x = 2 y + 3$

Solve: $y = \frac{x - 3}{2}$

Check: $2 \cdot \frac{x - 3}{2} + 3 = x$

Example: $f (x) = x^{3} - 1$

Swap: $x = y^{3} - 1$

Solve: $y^{3} = x + 1$ , so $f^{- 1} (x) = 3 x + 1$

Common mistake

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

The $- 1$ here doesn't mean reciprocal. For $f (x) = 2 x + 3$ , the inverse is $\frac{x - 3}{2}$ , not $\frac{1}{2 x + 3}$ .

Try it yourself

$f (x) = 5 x - 2$
Show answer
$f^{- 1} (x) = \frac{x + 2}{5}$
$f (x) = \frac{x}{4} + 1$
Show answer
$f^{- 1} (x) = 4 (x - 1)$
$f (x) = x^{3} + 2$
Show answer
$f^{- 1} (x) = 3 x - 2$

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