f(x) = x/4 - 3, find the inverse Step by Step

Step by step

The question

Question: find the inverse of f(x) = x/4 − 3.

f(x) = x4 − 3

The inverse function goes back.

If f takes 12 to 0, the inverse takes 0 back to 12.

inverse = goes back

f: 12 → 0 f ⁻¹: 0 → 12

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = x4 − 3

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 = y4 − 3

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.

y4 − 3 = x

4 Remove the − 3

We want y by itself.

The −3 is in the way.

To remove −3, we add 3.

We do it on both sides.

y4 ~~− 3 + 3~~ = x + 3

They make zero.

Cross them out.

Now we have y/4 = x + 3.

y4 = x + 3

5 Remove the fraction

There is a fraction: over 4.

Times 4 removes it.

We do it on both sides.

~~4 ×~~ y4 = 4 × (x + 3)

4/4 is 1.

The fraction is gone.

So y = 4x + 12.

y = 4x + 12

6 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = 4x + 12

The little −1 is part of the name.

It is not a power.

the −1 is a name,

not a power

7 Check

Let's check with a number.

Put 12 into f.

We get 0.

f(12) = 124 − 3 = 3 − 3 = 0

Now put 0 into the inverse.

We get 12.

f ⁻¹(0) = 4(0) + 12 = 0 + 12 = 12

The inverse took 0 back to 12.

So f⁻¹(x) = 4x + 12 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: remove the fraction.

Then: write the inverse.

Last: check.

Answer