Inverse of f(x) = x^2 + 3, x >= 0 Easy

The question

Question: find the inverse of f(x) = x² + 3, x ≥ 0.

The inverse goes back: what comes out of f goes in, and what went in comes out.

1 Why x ≥ 0

Without the rule x ≥ 0, two x's give the same answer.

f(2) = 7...

...and f(−2) = 7 too.

Going back from 7, which one?

There would be two ways back.

With x ≥ 0, only 2 is allowed.

Now each answer has one way back.

2 Write y =

First we write y for f(x): y = x² + 3.

3 Swap x and y

Swap x and y: what comes out now goes in.

x = y² + 3.

4 Get y alone

To remove + 3, we take away 3.

We do it on both sides, so they stay equal.

+ 3 and − 3 make zero.

Cross them out.

Now we have x − 3 = y².

To remove the square, we take the square root of both sides.

y is f's x, so y is 0 or more.

So only the plus root.

Careful!

Not plus or minus!

The inverse must give one answer.

y comes from f's x's, which are 0 or more: the plus root.

So √(x − 3) = y.

y is alone: y = √(x − 3).

This y is the inverse: f⁻¹(x) = √(x − 3).

5 The domain of the inverse

The inverse's x's are f's answers.

f(x) is a square plus 3, so it is 3 or more.

So the inverse only takes x ≥ 3.

6 Draw both

Draw the grid.

Plot points of f: (0, 3), (1, 4) and (2, 7).

Join them: f(x) = x² + 3, starting at (0, 3).

For the inverse, swap each point: (3, 0), (4, 1) and (7, 2).

Join them: f⁻¹, starting at (3, 0).

The two graphs are mirror images in the line y = x.

7 Check

Let's check: there and back.

f(2) = 7.

Now put 7 into the inverse.

We get back 2.

It works!

Let's look back.

First: why x ≥ 0.

Then: write y =.

Then: swap x and y.

Then: get y alone.

Then: the domain of the inverse.

Then: draw both.

Last: check.

▶ Watch it solved

Inverse of f(x) = (x - 2)^2, x >= 2 Medium

The question

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

The inverse goes back: what comes out of f goes in, and what went in comes out.

1 Why x ≥ 2

Without the rule x ≥ 2, two x's give the same answer.

f(4) = 4...

...and f(0) = 4 too.

Going back from 4, which one?

There would be two ways back.

With x ≥ 2, only 4 is allowed.

Now each answer has one way back.

2 Write y =

First we write y for f(x): y = (x − 2)².

3 Swap x and y

Swap x and y: what comes out now goes in.

x = (y − 2)².

4 Get y alone

To remove the square, we take the square root of both sides.

y is f's x, so y is at least 2.

So only the plus root.

Careful!

Not plus or minus!

The inverse must give one answer.

y comes from f's x's, which are at least 2: the plus root.

So √(x) = y − 2.

To remove − 2, we add 2.

We do it on both sides, so they stay equal.

− 2 and + 2 make zero.

Cross them out.

y is alone: y = √x + 2.

This y is the inverse: f⁻¹(x) = √x + 2.

5 The domain of the inverse

The inverse's x's are f's answers.

f(x) is a square , so it is 0 or more.

So the inverse only takes x ≥ 0.

6 Draw both

Draw the grid.

Plot points of f: (2, 0), (3, 1) and (4, 4).

Join them: f(x) = (x − 2)², starting at (2, 0).

For the inverse, swap each point: (0, 2), (1, 3) and (4, 4).

Join them: f⁻¹, starting at (0, 2).

The two graphs are mirror images in the line y = x.

7 Check

Let's check: there and back.

f(4) = 4.

Now put 4 into the inverse.

We get back 4.

It works!

Let's look back.

First: why x ≥ 2.

Then: write y =.

Then: swap x and y.

Then: get y alone.

Then: the domain of the inverse.

Then: draw both.

Last: check.

▶ Watch it solved

Inverse of f(x) = (x - 2)^2 + 3, x >= 2 Harder

The question

Question: find the inverse of f(x) = (x − 2)² + 3, x ≥ 2.

The inverse goes back: what comes out of f goes in, and what went in comes out.

1 Why x ≥ 2

Without the rule x ≥ 2, two x's give the same answer.

f(4) = 7...

...and f(0) = 7 too.

Going back from 7, which one?

There would be two ways back.

With x ≥ 2, only 4 is allowed.

Now each answer has one way back.

2 Write y =

First we write y for f(x): y = (x − 2)² + 3.

3 Swap x and y

Swap x and y: what comes out now goes in.

x = (y − 2)² + 3.

4 Get y alone

To remove + 3, we take away 3.

We do it on both sides, so they stay equal.

+ 3 and − 3 make zero.

Cross them out.

Now we have x − 3 = (y − 2)².

To remove the square, we take the square root of both sides.

y is f's x, so y is at least 2.

So only the plus root.

Careful!

Not plus or minus!

The inverse must give one answer.

y comes from f's x's, which are at least 2: the plus root.

So √(x − 3) = y − 2.

To remove − 2, we add 2.

We do it on both sides, so they stay equal.

− 2 and + 2 make zero.

Cross them out.

y is alone: y = √(x − 3) + 2.

This y is the inverse: f⁻¹(x) = √(x − 3) + 2.

5 The domain of the inverse

The inverse's x's are f's answers.

f(x) is a square plus 3, so it is 3 or more.

So the inverse only takes x ≥ 3.

6 Draw both

Draw the grid.

Plot points of f: (2, 3), (3, 4) and (4, 7).

Join them: f(x) = (x − 2)² + 3, starting at (2, 3).

For the inverse, swap each point: (3, 2), (4, 3) and (7, 4).

Join them: f⁻¹, starting at (3, 2).

The two graphs are mirror images in the line y = x.

7 Check

Let's check: there and back.

f(4) = 7.

Now put 7 into the inverse.

We get back 4.

It works!

Let's look back.

First: why x ≥ 2.

Then: write y =.

Then: swap x and y.

Then: get y alone.

Then: the domain of the inverse.

Then: draw both.

Last: check.

▶ Watch it solved