Solve f(x) = x/2 + 1, f^-1(x) Step by Step

Step by step

The question

Question: find the inverse of f(x) = x/2 + 1.

f(x) = x2 + 1

The inverse function goes back.

If f takes 4 to 3, the inverse takes 3 back to 4.

inverse = goes back

f: 4 → 3 f ⁻¹: 3 → 4

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = x2 + 1

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 = y2 + 1

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.

y2 + 1 = x

4 Remove the + 1

We want y by itself.

The 1 is in the way.

To remove 1, we take away 1.

We do it on both sides.

y2 ~~+ 1 − 1~~ = x − 1

They make zero.

Cross them out.

Now we have y/2 = x − 1.

y2 = x − 1

5 Remove the fraction

There is a fraction: over 2.

Times 2 removes it.

We do it on both sides.

~~2 ×~~ y2 = 2 × (x − 1)

2/2 is 1.

The fraction is gone.

So y = 2x − 2.

y = 2x − 2

6 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = 2x − 2

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 4 into f.

We get 3.

f(4) = 42 + 1 = 2 + 1 = 3

Now put 3 into the inverse.

We get 4.

f ⁻¹(3) = 2(3) − 2 = 6 − 2 = 4

The inverse took 3 back to 4.

So f⁻¹(x) = 2x − 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 + 1.

Then: remove the fraction.

Then: write the inverse.

Last: check.

Answer