f(x) = x - 9, find the inverse Step by Step

Step by step

The question

Question: find the inverse of f(x) = x − 9.

f(x) = x − 9

The inverse function goes back.

If f takes 10 to 1, the inverse takes 1 back to 10.

inverse = goes back

f: 10 → 1 f ⁻¹: 1 → 10

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = x − 9

2 Swap x and y

The inverse goes back: the answer goes in, and the start comes out.

So x and y swap places.

x = y − 9

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.

y − 9 = x

4 Remove the − 9

We want y by itself.

The −9 is in the way.

To remove −9, we add 9.

We do it on both sides.

y ~~− 9 + 9~~ = x + 9

They make zero.

Cross them out.

So y = x + 9.

y = x + 9

5 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = x + 9

The little −1 is part of the name.

It is not a power.

the −1 is a name,

not a power

6 Check

Let's check with a number.

Put 10 into f.

We get 1.

f(10) = 10 − 9 = 1

Now put 1 into the inverse.

We get 10.

f ⁻¹(1) = 1 + 9 = 10

The inverse took 1 back to 10.

So f⁻¹(x) = x + 9 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 − 9.

Then: write the inverse.

Last: check.

Answer