Step by step
The question
Question: find the inverse of f(x) = 2(x − 1)³ − 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 = 2(x − 1)³ − 3.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = 2(y − 1)³ − 3.
3 Get y alone
Two things are stuck to the bracket: times 2 and the − 3.
The outside one comes off first.
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 = 2(y − 1)³.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just the bracket is left.
Now we have (x + 3)/2 = (y − 1)³.
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 + 3)/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 + 3)/2) + 1.
This y is the inverse: f⁻¹(x) = ∛((x + 3)/2) + 1.
4 Draw both
Draw the grid.
Plot points of f: (0, −5), (1, −3) and (2, −1).
Join them: f(x) = 2(x − 1)³ − 3.
For the inverse, swap each point: (−5, 0), (−3, 1) and (−1, 2).
Join them: f⁻¹.
The two graphs are mirror images in the line y = x.
5 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: draw both.
Last: check.