Step by step
The question
Question: find the inverse of f(x) = (x + 1)² − 2, x ≥ −1.
The inverse goes back: what comes out of f goes in, and what went in comes out.
1 Why x ≥ −1
Without the rule x ≥ −1, two x's give the same answer.
f(1) = 2...
...and f(−3) = 2 too.
Going back from 2, which one?
There would be two ways back.
With x ≥ −1, only 1 is allowed.
Now each answer has one way back.
2 Write y =
First we write y for f(x): y = (x + 1)² − 2.
3 Swap x and y
Swap x and y: what comes out now goes in.
x = (y + 1)² − 2.
4 Get y alone
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
Now we have x + 2 = (y + 1)².
To remove the square, we take the square root of both sides.
y is f's x, so y is at least −1.
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 at least −1: the plus root.
So √(x + 2) = y + 1.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
y is alone: y = √(x + 2) − 1.
This y is the inverse: f⁻¹(x) = √(x + 2) − 1.
5 The domain of the inverse
The inverse's x's are f's answers.
f(x) is a square minus 2, so it is −2 or more.
So the inverse only takes x ≥ −2.
6 Draw both
Draw the grid.
Plot points of f: (−1, −2), (0, −1) and (1, 2).
Join them: f(x) = (x + 1)² − 2, starting at (−1, −2).
For the inverse, swap each point: (−2, −1), (−1, 0) and (2, 1).
Join them: f⁻¹, starting at (−2, −1).
The two graphs are mirror images in the line y = x.
7 Check
Let's check: there and back.
f(1) = 2.
Now put 2 into the inverse.
We get back 1.
It works!
Let's look back.
First: why x ≥ −1.
Then: write y =.
Then: swap x and y.
Then: get y alone.
Then: the domain of the inverse.
Then: draw both.
Last: check.