The inverse of y = x^2

Question

 Explain why the inverse of a function is not always a function?

Answer 1

The inverse of f(x) = x² is f^-1(x) = x^1/2.

But f^-1(x) has 2 values for each x, namely x and -x; therefore it is not a function.

In order for the inverse of a function to be a function it must be either non-increasing or non-decreasing on the whole real line.  Use your graphing calculator to graph y=1-e^-x and ln(1/(1-x)) to see what inverses look like.