Inverse of f(x) = 3x Step by Step | Composition and inverse functions

Step by step

The question

Question: find the inverse of f(x) = 3x.

f(x) = 3x

The inverse function goes back.

If f takes 2 to 6, the inverse takes 6 back to 2.

inverse = goes back

f: 2 → 6 f ⁻¹: 6 → 2

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = 3x

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 = 3y

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.

3y = x

4 Remove the 3

3y means 3 times y.

We want just y.

To remove times 3, we divide by 3.

We do it on both sides.

3y3 = x3

3 divided by 3 is 1.

So just y is left.

So y = x/3.

y = x3

5 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = x3

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 2 into f.

We get 6.

f(2) = 3(2) = 6

Now put 6 into the inverse.

We get 2.

f ⁻¹(6) = 63 = 2

The inverse took 6 back to 2.

So f⁻¹(x) = x/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 3.

Then: write the inverse.

Last: check.

Answer