The graph of f(x)=x^2 - 2 is that of a parabola that opens up and has its vertex at (0, -2). A horizontal test line drawn through this graph intersects the graph in 2 places, which indicates that f(x)=x^2 - 2 per se does not have an inverse function.
However, if we restrict the domain of f(x)=x^2 - 2 to [0, ∞), the graph is the right half of that of f(x)=x^2 - 2. This part of the graph shows that f(x)=x^2 - 2 on [0, ∞) has an inverse.
To find this inverse algebraically:
1) replace "f(x)" with "y": y = x^2 - 2
2) interchange x and y: x = y^2 - 2
3) solve this result for y: y^2 = x + 2, or y = +√(x + 2)
4) replace this 'y' with the symbol for 'inverse function of x:'
-1
f (x) = +√(x + 2). The domain of this function is (-∞, -2].
