Graphs of Inverse Functions

Video lesson · from 1:15:231 worked example

One lesson video covers all of Functions & Graphs, so it opens at 1:15:23 for this topic — not from the beginning.

If we have a graph of a function, we can find the graph of the inverse function by reflecting it in the line $y = x$ .

Every coordinate $(a,b)$ on the original graph becomes $(b,a)$ on the inverse graph.

$f(a) = b \implies f^{-1}(b) = a$

Reflect in $y = x$ : not in either axis. Every point $(a,b)$ becomes $(b,a)$ .
Draw the line $y = x$ on your sketch: it makes the reflection much easier to get right, and shows the marker what you did.
Domain and range swap over: the domain of the inverse is the range of the original, and vice versa.
Intercepts swap too: an $x$ -intercept of the original becomes a $y$ -intercept of the inverse.

Example 1

Shown is the graph of $f(x)$ .

On the same diagram, sketch its inverse $f^{-1}(x)$ .

To sketch the inverse, we reflect the curve in the line $y = x$ .

The coordinates of the indicated points will be reversed:

Plot these new points and draw a smooth curve through them, ensuring it is a perfect reflection across the line $y=x$ .