Inverse functions

7 minAS Level1.2.11.2.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can say what an inverse function is, find $f^{-1}(x)$ for a one-one function, pick the right sign when a square root appears, and state the domain of $f^{-1}$ .

Ctrl+Z for functions

A function takes an input and does something to it. Its inverse is the undo button: feed it the output, and it hands you back the input you started with.

$f(4) = 11$ means $f$ sends 4 to 11.
$f^{-1}(11) = 4$ means $f^{-1}$ sends 11 straight back to 4.

Only one-one functions get an undo button. If two inputs shared the output 9, then "undo 9" would have two answers, and a function can't do that.

Definition

Inverse function

If $f$ is a one-one function, its inverse function $f^{-1}$ maps each value in the range of $f$ back to the value in the domain it came from:

$\text{if } f(a) = b, \text{ then } f^{-1}(b) = a$

Read $f^{-1}$ as "f inverse". The $-1$ is a label, not a power.
The domain of $f^{-1}$ is the range of $f$ . The range of $f^{-1}$ is the domain of $f$ . Inputs and outputs swap roles.
Doing $f$ then $f^{-1}$ gets you back where you started: $f^{-1}f(x) = x$ . (Here $f^{-1}f(x)$ means "apply $f$ , then apply $f^{-1}$ to the answer".)
The other order works too: $ff^{-1}(x) = x$ (for $x$ in the range of $f$ ).

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.