Question

How do you find the inverse of `(x + 2)^2 - 4` and is it a function?

Answer 1

The inverse of a function is found algebraiccally by switching the x and y values.

Explanation:

`y = (x + 2)^2 - 4 -> x = (y + 2)^2 - 4`

`x + 4 = (y + 2)^2`

`+-sqrt(x + 4) = y + 2`

`+-sqrt(x + 4) - 2 = y`

`f^-1(x) = +-sqrt(x + 4) - 2`

This is not a function, because of the `+-` sign. For example, we can substitute x = 12 into the function to find y.

`y = sqrt(12 + 4) - 2`

or

`y = -sqrt(12 + 4) - 2`

`-> y = 2`

or

`-> y = -6`

Since the definition of a function is a relation where each x value has one and only one y value, and that this function contains the points `(12, 2) and (12, -6)`, this is not a function.

Hopefully you understand now!