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

Question

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

Lily Anderson · Accepted Answer

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

#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!

Cora Alexander · Answer

undefined To find the inverse of $ (x + 2)^2 - 4 $, follow these steps:

1. Replace $ f(x) $ with $ y $.
2. Swap the roles of $ x $ and $ y $.
3. Solve the resulting equation for $ y $ to get the inverse function.

So, starting with $ y = (x + 2)^2 - 4 $:

$$ x = (y + 2)^2 - 4 $$
$$ x + 4 = (y + 2)^2 $$
$$ \sqrt{x + 4} = y + 2 $$
$$ y = \sqrt{x + 4} - 2 $$

Therefore, the inverse of $ (x + 2)^2 - 4 $ is $ \sqrt{x + 4} - 2 $.

Regarding whether it's a function, yes, it is a function because for each input $ x $, there is exactly one output $ y $.