How do I find the inverse?

g(x) as given has no inverse because there are instances of two x values giving the same value of g(x). For instance,

x = -1   ⇒   g(-1) = 4 (-1 + 3)² - 8 = 8

x = -5   ⇒   g(-5) = 4 (-5 + 3)² - 8 = 8

Only a one-to-one function can have an inverse. g(x) is not one-to-one.

However, if we restrict the domain of g(x), we can find an inverse over that domain. Let be the inverse of g(x). Then by definition of inverse function,

Solve for the inverse:

Recall the definition of absolute value:

This means there are two possible solutions for the inverse of g(x) :

• if , then

• otherwise, if , then

Which we choose as the inverse depends on how we restrict the domain of g(x). For example:

Remember that the inverse must satisfy

In the first case above, , or . This suggests that we could restrict the domain of g(x) to be .

Then as long as , the inverse is

ab will be equal to