Inverse Functions

Part 4: Inverting the Tangent Function

1. From Part 3, we have the derivative formula
   

   Explain why the function G defined by the formula

   

   must be the arctangent function. What is the domain of this function? What is its range? How do you know?

2. Graph the functions G(x) and tan x together. Does G appear to have the right properties to be the inverse of the tangent function? Explain.

3. Graph the composite functions G(tan x) and tan(G(x)). Explain what you see. In particular, what is the domain of each of these functions? Why? [Warning: Your computer algebra system may not know the domains of these functions and may do something strange. You are responsible for a correct mathematical answer even if the computer result is misleading. You don't have to explain what the CAS is doing -- just explain the mathematics.]