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