The graph of f(x) (blue) and it's inverse (red) are shown. Each function has a different inverse for each of the intervals for which it is injective. You can change the function o the top of the left panel. You can type, for example:
x3 - 3x (not injective)
3x - 2 (linear function)
sen(x) (sine function)
sqrt(x) (square root of x)
exp(x) (exponential function, f(x) = ex)
You can drag the white point on the x.asis or animate it by clicking on the "play" button at the left bottom corner.
As you can see, the two graphs are symmetrical respect to the line y = x. Notice also that if f(a) = b, then f-1(b) = a. If a function is not injective, then it does not have ONE inverse function, but more than one: one for each of the intervals in which it is injective. In the case of f(x) = x3 - 3x, There are three inverse functions. One in (-∞, 1), another in (-1, 1) and another one in (-1, ∞). Can you see a relationship between the slopes of the tangent lines on corresponding points?