Composite and Inverse Functions: f(g(x)) and f⁻¹(x) Explained

Q: Does f(g(x)) equal g(f(x))?

Usually not. With f(x) = x^2 and g(x) = x + 3, f(g(x)) = (x + 3)^2 but g(f(x)) = x^2 + 3.

Q: How do you find an inverse function?

Write y = f(x), swap x and y, then solve for y. The result is f^-1(x).

How to chain functions together and why the order matters, how to find an inverse by swapping $x$ and $y$ , and why the graph of an inverse is a reflection in the line $y = x$ .

3 min read Supports 4 lessons

Composite functions

Functions can be chained like machines on a production line. The output of the first becomes the input of the second. Feeding $x$ into $g$ and then the result into $f$ gives the composite $f(g(x))$ , also written $(f \circ g)(x)$ .

x \;\xrightarrow{\;g\;}\; g(x) \;\xrightarrow{\;f\;}\; f\big(g(x)\big)

$f(x) = x^2, \quad g(x) = x + 3$
Find $f(g(2))$ and $g(f(2))$ .

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Watch out

Order matters. $f(g(x))$ means do $g$ first. Work from the inside out.

You can compose with expressions too: $f(g(x)) = (x + 3)^2$ while $g(f(x)) = x^2 + 3$ .

Inverse functions

An inverse function runs the machine backwards. If $f$ doubles a number and adds $1$ , its inverse subtracts $1$ and halves. Applying a function and then its inverse returns you to where you started: $f^{-1}(f(x)) = x$ .

Finding an inverse

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

Worked example

Find the inverse of $f(x) = 3x - 6$ .

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Write $y = 3x - 6$ and swap $x$ and $y$ .
$x = 3y - 6$

Tip

Check: $f(4) = 6$ and $f^{-1}(6) = \frac{12}{3} = 4$ ✓.

The graph of an inverse

Swapping $x$ and $y$ turns every point $(a, b)$ on the graph of $f$ into $(b, a)$ on the graph of $f^{-1}$ . Geometrically, that is a reflection in the line $y = x$ .

Mirror images in the line $y = x$ .

When there is no inverse

A function only has an inverse if each output comes from just one input. $f(x) = x^2$ sends both $3$ and $-3$ to $9$ , so "undo" would not know which to return. Restricting the domain to $x \ge 0$ fixes this, and then the inverse is $\sqrt{x}$ .

Watch out

$f^{-1}(x)$ is not $\frac{1}{f(x)}$ . The $-1$ here means "inverse function", not a power.

Key idea

Some of the most important functions are inverse pairs: squaring and square roots, exponentials and logarithms, sine and $\sin^{-1}$ .

Common questions

Does f(g(x)) equal g(f(x))?: Usually not. With $f(x) = x^2$ and $g(x) = x + 3$ , $f(g(x)) = (x + 3)^2$ but $g(f(x)) = x^2 + 3$ .
How do you find an inverse function?: Write $y = f(x)$ , swap $x$ and $y$ , then solve for $y$ . The result is $f^{-1}(x)$ .

Practise composite and inverse functions

The Composite Functions lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Does f(g(x)) equal g(f(x))?

Usually not. With $f(x) = x^2$ and $g(x) = x + 3$ , $f(g(x)) = (x + 3)^2$ but $g(f(x)) = x^2 + 3$ .

How do you find an inverse function?

Write $y = f(x)$ , swap $x$ and $y$ , then solve for $y$ . The result is $f^{-1}(x)$ .