Section 5.3 Inverse Functions

Recall the following definition regarding inverse functions: Functions f and g are inverses of each other if and only if and for all x in the domains of the individual functions. Additionally, the graphs of functions f and g are reflections of each other over the line y = x

Section 5.3 Inverse Functions How can we determine whether a function has an inverse? A function has an inverse which is a function only if it is a one-to-one function. Functions are one-to-one if and only if for each x there is only one y (the definition of a function) AND for each y there is only one x.

Section 5.3 Inverse Functions If we don’t have a complete graph, how can we tell if it is one-to-one? Well, Calculus can help. A graph can only be one-to-one if it never changes direction. This can only happen when there are no relative Max or min for the function, which we can determine through the use of a sign chart. In those cases we say that the function is strictly monotonic.

Section 5.3 Inverse Functions For each of the following functions, determine whether the function has an inverse. If it does, try to find that inverse:

Section 5.3 Inverse Functions The examples on the previous slide point out a couple of important facts about the study of inverse functions;  Most functions do not have inverses that are functions  Even when a function has an inverse function, it is a difficult task to solve for the inverse in our typical function notation

Section 5.3 Inverse Functions There is an interesting relationship between the derivative of a function and the derivative of the inverse of the function. The key idea to this relationship is to remind yourself that whenever the point (a,b) lies on the graph of a function, the point (b,a) is a point on the graph of the inverse. It stands to reason, then, that there should be a relationship between the derivative of f at a and the derivative of the inverse of f at b.

Section 5.3 Inverse Functions What is this relationship? We can describe it most clearly with the following definition: Let f be a differentiable function that has an inverse function g, then the following equation holds true: