How does an inverse function work here? What's an isolated y-variable?

Question

I'm given the quadratic function f(x) = x^2 - 2. With this function, one must find the coordinates of the vertex of x = f(y) and then determine the entire equation of the inverse relation with an isolated y-variable.

Answer 1

There are several approaches to finding the vertex.

I am not sure which one you are to use.

But in this case you can simply notice that the function is symmetrical

around the vertical axis, which puts the vertex at x = 0.

"Isolated y-variable" means y-variable treated as the independent variable,

and expressing x in terms of y.

You had the right idea when you wrote x = f(y), but it should have been

x = f^-1(y), where

f^-1 denotes the inverse function of f.

You can find the inverse function by starting with

y = x² - 2

and then solving for x:

x = √(y+2)

And there is a second inverse function:

x = -√(y+2)