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.[br]You can change the function o the top of the left panel. You can type, for example:[br][br][*] x[sup]3[/sup] - 3x (not injective)[br][/*][*] 3x - 2 (linear function)[br][/*][*] sen(x) (sine function)[br][/*][*] sqrt(x) (square root of x)[br][/*][*] exp(x) (exponential function, f(x) = e[sup]x[/sup])[br][br]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[sup]-1[/sup](b) = a.[br][br]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) = x[sup]3[/sup] - 3x, There are three inverse functions. One in (-∞, 1), another in (-1, 1) and another one in (-1, ∞).[br][br]Can you see a relationship between the slopes of the tangent lines on corresponding points?