Step by step
The question
Question: find the inverse of f(x) = 2x³ − 5.
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 = 2x³ − 5.
2 Swap x and y
Swap x and y: what comes out now goes in.
x = 2y³ − 5.
3 Get y alone
Two things are stuck to the bracket: times 2 and the − 5.
The outside one comes off first.
To remove − 5, we add 5.
We do it on both sides, so they stay equal.
− 5 and + 5 make zero.
Cross them out.
Now we have x + 5 = 2y³.
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 + 5)/2 = 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 + 5)/2 = y.
y is alone: y = ∛((x + 5)/2).
This y is the inverse: f⁻¹(x) = ∛((x + 5)/2).
4 Draw both
Draw the grid.
Plot points of f: (−1, −7), (0, −5) and (1, −3).
Join them: f(x) = 2x³ − 5.
For the inverse, swap each point: (−7, −1), (−5, 0) and (−3, 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) = −3.
Now put −3 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.