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