Solve f(x) = 7 - 2x, inverse Step by Step

Step by step

The question

Question: find the inverse of f(x) = 7 − 2x.

f(x) = 7 − 2x

The inverse function goes back.

If f takes 2 to 3, the inverse takes 3 back to 2.

inverse = goes back

f: 2 → 3 f ⁻¹: 3 → 2

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = 7 − 2x

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 = 7 − 2y

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.

7 − 2y = x

4 Remove the 7

We want y by itself.

The 7 is in the way.

To remove 7, we take away 7.

We do it on both sides.

7 − 2y ~~− 7~~ = x − 7

They make zero.

Cross them out.

Now we have −2y = x − 7.

−2y = x − 7

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.

−2y−2 = x − 7−2

−2 divided by −2 is 1.

So just y is left.

Now we have y = (x − 7) over −2.

y = x − 7−2

6 Make it simpler

A minus at the bottom is hard to read.

We change the sign of the top and the bottom.

The fraction stays the same.

−A−B = AB

So y = (7 − x)/2.

y = 7 − x2

7 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = 7 − x2

The little −1 is part of the name.

It is not a power.

the −1 is a name,

not a power

8 Check

Let's check with a number.

Put 2 into f.

We get 3.

f(2) = 7 − 2(2) = 7 − 4 = 3

Now put 3 into the inverse.

We get 2.

f ⁻¹(3) = 7 − 32 = 42 = 2

The inverse took 3 back to 2.

So f⁻¹(x) = (7 − x)/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 7.

Then: remove the −2.

Then: make it simpler.

Then: write the inverse.

Last: check.

Answer