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