Step by step
The question
Question: find the inverse of f(x) = x² − 1, x ≥ 0.
The inverse goes back: what comes out of f goes in, and what went in comes out.
1 Why x ≥ 0
Without the rule x ≥ 0, two x's give the same answer.
f(2) = 3...
...and f(−2) = 3 too.
Going back from 3, which one?
There would be two ways back.
With x ≥ 0, only 2 is allowed.
Now each answer has one way back.
2 Write y =
First we write y for f(x): y = x² − 1.
3 Swap x and y
Swap x and y: what comes out now goes in.
x = y² − 1.
4 Get y alone
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
Now we have x + 1 = y².
To remove the square, we take the square root of both sides.
y is f's x, so y is 0 or more.
So only the plus root.
Careful!
Not plus or minus!
The inverse must give one answer.
y comes from f's x's, which are 0 or more: the plus root.
So √(x + 1) = y.
y is alone: y = √(x + 1).
This y is the inverse: f⁻¹(x) = √(x + 1).
5 The domain of the inverse
The inverse's x's are f's answers.
f(x) is a square minus 1, so it is −1 or more.
So the inverse only takes x ≥ −1.
6 Draw both
Draw the grid.
Plot points of f: (0, −1), (1, 0) and (2, 3).
Join them: f(x) = x² − 1, starting at (0, −1).
For the inverse, swap each point: (−1, 0), (0, 1) and (3, 2).
Join them: f⁻¹, starting at (−1, 0).
The two graphs are mirror images in the line y = x.
7 Check
Let's check: there and back.
f(2) = 3.
Now put 3 into the inverse.
We get back 2.
It works!
Let's look back.
First: why x ≥ 0.
Then: write y =.
Then: swap x and y.
Then: get y alone.
Then: the domain of the inverse.
Then: draw both.
Last: check.