Step by step
The question
Question: find the inverse of f(x) = (x + 1)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 + 1)³.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = (y + 1)³.
3 Get y alone
To remove the cube, we take the cube root of both sides.
A cube root has just one answer: 2 × 2 × 2 = 8, so the cube root of 8 is 2.
Careful!
A cube root has only one answer.
Minus 2 cubed is minus 8, not 8.
So no plus or minus here.
So ∛x = y + 1.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
y is alone: y = ∛x − 1.
This y is the inverse: f⁻¹(x) = ∛x − 1.
4 Draw both
Draw the grid.
Plot points of f: (−2, −1), (−1, 0) and (0, 1).
Join them: f(x) = (x + 1)³.
For the inverse, swap each point: (−1, −2), (0, −1) and (1, 0).
Join them: f⁻¹.
The two graphs are mirror images in the line y = x.
5 Check
Let's check: there and back.
f(0) = 1.
Now put 1 into the inverse.
We get back 0.
It works!
Let's look back.
First: write y =.
Then: swap x and y.
Then: get y alone.
Then: draw both.
Last: check.