Inverse of f(x) = √(x + 4)

Step by step

The question

Question: find the inverse of f(x) = √(x + 4) − 1.

f(x) = √x + 4 − 1

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

inverse: swaps in and out

1 Write y =

First we write y for f(x): y = √(x + 4) − 1.

y = √x + 4 − 1

2 Swap x and y

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

x = √(y + 4) − 1.

x = √y + 4 − 1

3 Get y alone

To remove − 1, we add 1.

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

x + 1 = √y + 4 ~~− 1 + 1~~

− 1 and + 1 make zero.

Cross them out.

Now we have x + 1 = √(y + 4).

x + 1 = √y + 4

To remove the square root, we square both sides.

Squaring removes the square root.

(√9)² = 9: squaring removes the √

Square both sides: (x + 1)² = y + 4.

(x + 1)² = y + 4

To remove + 4, we take away 4.

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

(x + 1)² − 4 = ~~y + 4 − 4~~

+ 4 and − 4 make zero.

Cross them out.

y is alone: y = (x + 1)² − 4.

y = (x + 1)² − 4

This y is the inverse: f⁻¹(x) = (x + 1)² − 4.

f ⁻¹(x) = (x + 1)² − 4, x ≥ −1

4 The domain of the inverse

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

f(x) is a square root minus 1, so it is −1 or more.

f's y's = the inverse's x's

Careful!

Without it, the inverse would give answers f never had.

Its x's are only f's answers: −1 or more.

Careful: keep x ≥ −1 for the inverse

So the inverse only takes x ≥ −1.

x ≥ −1

5 Draw both

Draw the grid.

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

Join them: f(x) = √(x + 4) − 1, starting at (−4, −1).

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

Join them: f⁻¹, starting at (−1, −4).

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

6 Check

Let's check: there and back.

f(0) = 1.

f(0) = √0 + 4 − 1 = 1

Now put 1 into the inverse.

We get back 0.

It works!

f ⁻¹(1) = (1 + 1)² − 4 = (2)² − 4 = 0

Let's look back.

First: write y =.

Then: swap x and y.

Then: get y alone.

Then: the domain of the inverse.

Then: draw both.

Last: check.

Answer