Chapter 0.6, Problem 26E

To determine

To show: the given function is one-to-one, and to graph its inverse.

Answer to Problem 26E

The given function is one-to-one. The graph of its inverse is,

  

Explanation of Solution

Given:

  $y=\tan x,-\frac{\pi }{2}<x<\frac{\pi }{2}$ .

Calculation:

The graph of the restricted tangent function using the parametric equations $x_1=t,y_1=\tan t,-\frac{\pi }{2}<t<\frac{\pi }{2}$ is,

  

This restricted tangent function is one-to-one because it does not repeat any output values.

So it has an inverse.

Now the graph of inverse is obtained by interchanging the ordered pairs using the parametric equations $x_2=\tan t,y_2=t,-\frac{\pi }{2}<t<\frac{\pi }{2}$ . That is,