Step by step
The question
Question: find the inverse of f(x) = 3x − 2.
The inverse function goes back.
We pick a small number that gives a whole answer.
For example, f takes 2 to 4.
The inverse takes 4 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.
Look at our numbers.
For f, x is 2 and y is 4.
For the inverse, x is 4 and y is 2.
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.
Is y alone now?
Not yet: − 2 is still with y.
How do we remove it?
4 Remove the − 2
We want y by itself.
The −2 is in the way.
To remove −2, we add 2.
We do it on both sides.
−2 and + 2 cancel each other.
They make zero.
Now we have 3y = x + 2.
So after step 4: we added 2, and now we have 3y = x + 2.
Is y alone now?
Not yet: 3y means 3 times y.
5 Remove the 3
To remove times 3, we divide by 3.
We do it on both sides.
3 divided by 3 is 1.
They cancel each other.
So just y is left.
So y = (x + 2)/3.
So after step 5: we divided by 3, and now we have y = x + 2/3.
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 4.
Now put 4 into the inverse.
We get 2.
The inverse took 4 back to 2.
So f⁻¹(x) = (x + 2)/3 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 − 2.
Then: remove the 3.
Then: write the inverse.
Last: check.