PrecalculusPrecalculusHigh school

Inverse Functions

Swap x and y, solve, and check both compositions.

An inverse function reverses another: if $f$ sends 3 to 10, then $f^{-1}$ sends 10 back to 3. Composing them in either order returns the input, $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ . Graphically, the inverse is the reflection of the original graph across the line $y = x$ .

Only one-to-one functions have inverses, because each output must come from exactly one input. The horizontal line test checks this. Functions that fail, like $x^2$ , can be given an inverse by restricting the domain, which is exactly how $\sqrt{x}$ and $\sin^{-1} x$ are defined.

Key terms

Inverse check: f and g are inverses when each undoes the other.
$f(g(x)) = g(f(x)) = x$

Key ideas

To find $f^{-1}$ : write $y = f(x)$ , swap $x$ and $y$ , solve for $y$ .
The domain of $f^{-1}$ is the range of $f$ , and vice versa.
$f^{-1}(x)$ is not $\frac{1}{f(x)}$ .
Verify by composing both ways.

Worked example

Find the inverse of $f(x) = \frac{2x - 1}{3}$ .

Write $y = \frac{2x - 1}{3}$ .
Swap: $x = \frac{2y - 1}{3}$ .
Solve: $3x = 2y - 1$ , so $y = \frac{3x + 1}{2}$ .
Check: $f(f^{-1}(x)) = \frac{(3x + 1) - 1}{3} = x$ .

Answer: $f^{-1}(x) = \frac{3x + 1}{2}$

Common mistakes

Reading $f^{-1}$ as a reciprocal.
Forgetting to restrict the domain of a function that is not one-to-one.
Solving for $x$ without swapping, which only rearranges the original.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.If $f(x) = 4x + 7$ , find $f^{-1}(15)$ .
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$2$
2.If $f(x) = x^3$ , find $f^{-1}(-27)$ .
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$-3$
3.If $f(x) = e^x$ , find $f^{-1}(1)$ .
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$0$

Keep practising

Calculators

Inverse Function Calculator