Inverse of f(x) = (x + 3)/4 Step by Step

Step by step

The question

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

f(x) = x + 34

The inverse function goes back.

If f takes 5 to 2, the inverse takes 2 back to 5.

inverse = goes back

f: 5 → 2 f ⁻¹: 2 → 5

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = x + 34

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 = y + 34

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.

y + 34 = x

4 Remove the fraction

There is a fraction: over 4.

Times 4 removes it.

We do it on both sides.

~~4 ×~~ y + 34 = 4 × x

4/4 is 1.

The fraction is gone.

Now we have y + 3 = 4x.

y + 3 = 4x

5 Remove the + 3

We want y by itself.

The 3 is in the way.

To remove 3, we take away 3.

We do it on both sides.

y ~~+ 3 − 3~~ = 4x − 3

They make zero.

Cross them out.

So y = 4x − 3.

y = 4x − 3

6 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = 4x − 3

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 5 into f.

We get 2.

f(5) = 54 + 34 = 2

Now put 2 into the inverse.

We get 5.

f ⁻¹(2) = 4(2) − 3 = 8 − 3 = 5

The inverse took 2 back to 5.

So f⁻¹(x) = 4x − 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 fraction.

Then: remove the + 3.

Then: write the inverse.

Last: check.

Answer