Section Summary: 7.1 - Inverse functions

  1. Definitions

  2. Theorems

    Differentiability is really about smoothness, and this too is reflected in the mirror of the line y=x. The only problem that can arise is if the slope of f is 0 at a point, because this will be reflected as an infinite slope of the inverse function (alternatively, an infinite slope of f turns into a 0 slope of tex2html_wrap_inline179 ).

  3. Properties/Tricks/Hints/Etc.

  4. Summary

    In this section we introduce the inverse function, which is the mirror image of f about the line y=x. Hence, properties of tex2html_wrap_inline179 are also mirror reflections of those properties of f. It's easy to graph tex2html_wrap_inline179 , for example, once we have the graph of f, and properties of continuity and differentiability are also easy to deduce. Even asymptotes are reflected! It's even easy to compute slopes at a point on the inverse function if we know the derivative of the function f.

    One can think of f and tex2html_wrap_inline179 as ways of passing between two sets (the domain and range of f, which are the range and domain of tex2html_wrap_inline179 ). A secret code is an example of a function and its inverse: if I give you a secret message, it's tranformed from an ordinary expression (the domain) by a function and passed into a strange form (in the range). The secret agent on the other end needs a method (the inverse function) that turns strange forms into ordinary expressions.

Problems:

Problems, pp. 414-416, #2-10, 12, 14, 16, 22, 23, 34, 38,

In class: #7-10, 13, 15, 24, 33, 36





LONG ANDREW E
Tue Apr 17 12:06:42 EDT 2001