Step by step
The question
Question: find the inverse of f(x) = −x³ + 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 = −x³ + 2.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = −y³ + 2.
3 Get y alone
Two things are stuck to the bracket: times −1 and the + 2.
The outside one comes off first.
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 = −y³.
To remove times −1, we divide by −1.
We do it on both sides, so they stay equal.
−1 divided by −1 is 1.
So just the bracket is left.
Now we have (x − 2) over −1 = y³.
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 − 2) over −1) = y.
y is alone: y = ∛((x − 2) over −1).
This y is the inverse: f⁻¹(x) = ∛((x − 2) over −1).
4 Draw both
Draw the grid.
Plot points of f: (−1, 3), (0, 2) and (1, 1).
Join them: f(x) = −x³ + 2.
For the inverse, swap each point: (3, −1), (2, 0) and (1, 1).
Join them: f⁻¹.
The two graphs are mirror images in the line y = x.
5 Check
Let's check: there and back.
f(1) = 1.
Now put 1 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: draw both.
Last: check.