Inverse of f(x) = 2x + 6 Step by Step

Step by step

The question

Question: find the inverse of f(x) = 2x + 6.

f(x) = 2x + 6

The inverse function goes back.

If f takes 2 to 10, the inverse takes 10 back to 2.

inverse = goes back

f: 2 → 10 f ⁻¹: 10 → 2

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = 2x + 6

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 = 2y + 6

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.

2y + 6 = x

4 Remove the + 6

We want y by itself.

The 6 is in the way.

To remove 6, we take away 6.

We do it on both sides.

2y ~~+ 6 − 6~~ = x − 6

They make zero.

Cross them out.

Now we have 2y = x − 6.

2y = x − 6

5 Remove the 2

2y means 2 times y.

We want just y.

To remove times 2, we divide by 2.

We do it on both sides.

2y2 = x − 62

2 divided by 2 is 1.

So just y is left.

So y = (x − 6)/2.

y = x − 62

6 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = x − 62

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 2 into f.

We get 10.

f(2) = 2(2) + 6 = 4 + 6 = 10

Now put 10 into the inverse.

We get 2.

f ⁻¹(10) = 10 − 62 = 42 = 2

The inverse took 10 back to 2.

So f⁻¹(x) = (x − 6)/2 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 + 6.

Then: remove the 2.

Then: write the inverse.

Last: check.

Answer