Step by step
The question
Question: find the inverse of f(x) = 5x + 4.
The inverse function goes back.
If f takes 2 to 14, the inverse takes 14 back to 2.
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 + 4
We want y by itself.
The 4 is in the way.
To remove 4, we take away 4.
We do it on both sides.
They make zero.
Cross them out.
Now we have 5y = x − 4.
5 Remove the 5
5y means 5 times y.
We want just y.
To remove times 5, we divide by 5.
We do it on both sides.
5 divided by 5 is 1.
So just y is left.
So y = (x − 4)/5.
6 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.
7 Check
Let's check with a number.
Put 2 into f.
We get 14.
Now put 14 into the inverse.
We get 2.
The inverse took 14 back to 2.
So f⁻¹(x) = (x − 4)/5 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 + 4.
Then: remove the 5.
Then: write the inverse.
Last: check.