Step by step
The question
Question: find the inverse of f(x) = √(x − 1) + 2.
The inverse goes back: what comes out of f goes in, and what went in comes out.
1 Write y =
First we write y for f(x): y = √(x − 1) + 2.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = √(y − 1) + 2.
3 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).
To remove the square root, we square both sides.
Squaring removes the square root.
Square both sides: (x − 2)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)² + 1.
This y is the inverse: f⁻¹(x) = (x − 2)² + 1.
4 The domain of the inverse
The inverse's x's are f's answers.
f(x) is a square root plus 2, so it is 2 or more.
Careful!
Without it, the inverse would give answers f never had.
Its x's are only f's answers: 2 or more.
So the inverse only takes x ≥ 2.
5 Draw both
Draw the grid.
Plot points of f: (1, 2), (2, 3) and (5, 4).
Join them: f(x) = √(x − 1) + 2, starting at (1, 2).
For the inverse, swap each point: (2, 1), (3, 2) and (4, 5).
Join them: f⁻¹, starting at (2, 1).
The two graphs are mirror images in the line y = x.
6 Check
Let's check: there and back.
f(5) = 4.
Now put 4 into the inverse.
We get back 5.
It works!
Let's look back.
First: write y =.
Then: swap x and y.
Then: get y alone.
Then: the domain of the inverse.
Then: draw both.
Last: check.