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(3) = 6...
...and f(−1) = 6 too.
Going back from 6, which one?
There would be two ways back.
With x ≥ 1, only 3 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 take away 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 add 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 plus 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), (2, 3) and (3, 6).
Join them: f(x) = (x − 1)² + 2, starting at (1, 2).
For the inverse, swap each point: (2, 1), (3, 2) and (6, 3).
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(3) = 6.
Now put 6 into the inverse.
We get back 3.
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.