Composite and inverse functions

How composite functions chain together (and why order matters), and how to find an inverse — what it does, how to build it, and the link to its graph.

Pure · Algebra & functions

Composite and inverse functions

How composite functions chain together (and why order matters), and how to find an inverse — what it does, how to build it, and the link to its graph.

Why it works

A function is a rule with an input and an output, plus a domain (allowed inputs) and range (resulting outputs).

Composite functions chain two rules.

fg(x)

means do $g$ first, then $f$ — read it inside-out, like nested brackets:

fg(x) = f(g(x))

. So if

f(x) = 2x + 1

and

g(x) = x^2

, then

fg(x) = f(x^2) = 2x^2 + 1

, whereas

gf(x) = g(2x + 1) = (2x + 1)^2

. Order matters —

fg \ne gf

in general.

Inverse functions undo the original:

f^{-1}

takes an output of

f

back to its input, so

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

. To find it:

write $y = f(x)$ ,
rearrange to make $x$ the subject,
swap the letters to write $f^{-1}(x)$ .

For

f(x) = 3x - 2

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

, so

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

— it reverses "

\times 3

then

-2

" into "

+2

then

\div 3

".

Two facts that follow:

The graph of $y = f^{-1}(x)$ is the **reflection of $y = f(x)$ in the line

y = x

** (swapping

x

and

y

swaps the axes).

The domain and range swap: domain of $f^{-1}$ = range of $f$ . And a function

only has an inverse if it is one-to-one (each output comes from exactly one input) — otherwise the reverse rule isn't a function.

(Beware:

f^{-1}

is the inverse, not the reciprocal

\frac{1}{f(x)}