Question

Find the f^-1(x) and it’s domain

Answer 1

`f^(-1)(x) = (x + 8)^2`

`x \ge -8`

Explanation:

Given

`f(x) = \sqrt x - 8`

Solving (a):
`f^(-1)(x)`

We have:

`f(x) = \sqrt x - 8`

Express f(x) as y

`y = \sqrt x - 8`

Swap x and y

`x = \sqrt y - 8`

Add 8 to
`both\ sides`

`x + 8 = \sqrt y - 8 + 8`

`x + 8 = \sqrt y`

Square both sides

`(x + 8)^2 = y`

Rewrite as:

`y = (x + 8)^2`

Express y as:
`f^(-1)(x)`

`f^(-1)(x) = (x + 8)^2`

To determine the domain, we have:

The original function is
`f(x) = \sqrt x - 8`

The range of this is:
`f(x) \ge -8`

The
`domain` of the
`inverse` function is the
`range` of the
`original` function.

Hence, the domain is:

`x \ge -8`