Step by step
The question
Question: find the inverse of f(x) = √(x + 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 + 3).
2 Swap x and y
Swap x and y: what comes out now goes in.
x = √(y + 3).
3 Get y alone
To remove the square root, we square both sides.
Squaring removes the square root.
Square both sides: (x)² = y + 3.
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.
y is alone: y = x² − 3.
This y is the inverse: f⁻¹(x) = x² − 3.
4 The domain of the inverse
The inverse's x's are f's answers.
f(x) is a square root , so it is 0 or more.
Careful!
Without it, the inverse would give answers f never had.
Its x's are only f's answers: 0 or more.
So the inverse only takes x ≥ 0.
5 Draw both
Draw the grid.
Plot points of f: (−3, 0), (−2, 1) and (1, 2).
Join them: f(x) = √(x + 3), starting at (−3, 0).
For the inverse, swap each point: (0, −3), (1, −2) and (2, 1).
Join them: f⁻¹, starting at (0, −3).
The two graphs are mirror images in the line y = x.
6 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: write y =.
Then: swap x and y.
Then: get y alone.
Then: the domain of the inverse.
Then: draw both.
Last: check.