Step by step
The question
Question: find the inverse of f(x) = 6 − x/3.
The inverse function goes back.
If f takes 3 to 5, the inverse takes 5 back to 3.
1 Write y for f(x)
f(x) is the answer that comes out.
Let's call it y.
2 Swap x and y
The inverse goes back: the answer goes in, and the start comes out.
So x and y swap places.
Now we find y again, like in any formula.
3 y on the left
y is on the right.
We swap the two sides, so y is on the left.
They are still equal.
4 Remove the 6
We want y by itself.
The 6 is in the way.
To remove 6, we take away 6.
We do it on both sides.
They make zero.
Cross them out.
Now we have −y/3 = x − 6.
5 Remove the fraction
There is a fraction: over 3.
Times 3 removes it.
We do it on both sides.
3/3 is 1.
The fraction is gone.
Now we have −y = 3x − 18.
6 Remove the −1
−1y means −1 times y.
We want just y.
To remove times −1, we divide by −1.
We do it on both sides.
−1 divided by −1 is 1.
So just y is left.
Now we have y = (3x − 18) over −1.
7 Make it simpler
We can make it simpler: we divide each part by −1.
So y = −3x + 18.
8 Write the inverse
This y is the inverse.
We write it f⁻¹(x).
The little −1 is part of the name.
It is not a power.
9 Check
Let's check with a number.
Put 3 into f.
We get 5.
Now put 5 into the inverse.
We get 3.
The inverse took 5 back to 3.
So f⁻¹(x) = −3x + 18 is right!
Let's look back.
First: write y for f(x).
Then: swap x and y.
Then: y on the left.
Then: remove the 6.
Then: remove the fraction.
Then: remove the −1.
Then: make it simpler.
Then: write the inverse.
Last: check.