h(x) = 3x − 4. Find the inverse function h⁻¹(x). Step by Step

Step by step

The question

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

f(x) = 3x − 4

The inverse function goes back.

We pick a small number that gives a whole answer.

For example, f takes 3 to 5.

The inverse takes 5 back to 3.

inverse = goes back

f(3) = 3(3) − 4

= 9 − 4

= 5

f ⁻¹: 5 → 3

1 Write y for f(x)

f(x) is the answer that comes out.

Let's call it y.

y = 3x − 4

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 = 3y − 4

Look at our numbers.

For f, x is 3 and y is 5.

For the inverse, x is 5 and y is 3.

f: x = 3

y = 5

f ⁻¹: x = 5, y

= 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.

3y − 4 = x

Is y alone now?

Not yet: − 4 is still with y.

How do we remove it?

4 Remove the − 4

We want y by itself.

The −4 is in the way.

To remove −4, we add 4.

We do it on both sides.

3y ~~− 4 + 4~~ = x + 4

−4 and + 4 cancel each other.

They make zero.

Now we have 3y = x + 4.

3y = x + 4

So after step 4: we added 4, and now we have 3y = x + 4.

← step 4: added 4

Is y alone now?

Not yet: 3y means 3 times y.

5 Remove the 3

To remove times 3, we divide by 3.

We do it on both sides.

3y3 = x + 43

3 divided by 3 is 1.

They cancel each other.

So just y is left.

So y = (x + 4)/3.

y = x + 43

So after step 5: we divided by 3, and now we have y = x + 4/3.

← step 5: divided by 3

6 Write the inverse

This y is the inverse.

We write it f⁻¹(x).

f ⁻¹(x) = x + 43

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 3 into f.

We get 5.

f(3) = 3(3) − 4 = 9 − 4 = 5

Now put 5 into the inverse.

We get 3.

f ⁻¹(5) = 5 + 43 = 93 = 3

The inverse took 5 back to 3.

So f⁻¹(x) = (x + 4)/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 − 4.

Then: remove the 3.

Then: write the inverse.

Last: check.

Answer