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