Step by step
The question
Question: find the inverse of f(x) = (x − 2)³ + 1.
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 + 1.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = (y − 2)³ + 1.
3 Get y alone
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.
Now we have x − 1 = (y − 2)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 − 1) = y − 2.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
y is alone: y = ∛(x − 1) + 2.
This y is the inverse: f⁻¹(x) = ∛(x − 1) + 2.
4 Draw both
Draw the grid.
Plot points of f: (1, 0), (2, 1) and (3, 2).
Join them: f(x) = (x − 2)³ + 1.
For the inverse, swap each point: (0, 1), (1, 2) and (2, 3).
Join them: f⁻¹.
The two graphs are mirror images in the line y = x.
5 Check
Let's check: there and back.
f(3) = 2.
Now put 2 into the inverse.
We get back 3.
It works!
Let's look back.
First: write y =.
Then: swap x and y.
Then: get y alone.
Then: draw both.
Last: check.