Inverse Function

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?