# Find Domain

• Jun 11th 2006, 06:44 PM
Kiwigirl
Solve equations

1. Give the range of the function y = tan^-1 x

2. Give the value of tan^-1 5
• Jun 11th 2006, 06:56 PM
ThePerfectHacker
Quote:

Originally Posted by Kiwigirl

1. Give the range of the function y = tan^-1 x

There are two ways to do this problem. One way is the cowardly way they teach in school using a graph in that case look below to see that the range is,
$\displaystyle y\in (-\pi/2,\pi/2)$

The second way. The function $\displaystyle \tan^{-1}(x)$ is defined as the inverse function of the function,
$\displaystyle \tan x \mbox{ on } -\pi/2<x<\pi/2$. Now there is a rule that the range of the inverse function is the domain of the orignal function. Since the domain of the original function is $\displaystyle (-\pi/2,\pi/2)$ thus the range of $\displaystyle \tan^{-1}(x)$ is $\displaystyle (-\pi/2,\pi/2)$

There is actually a third way (much more advanced). Find all such $\displaystyle y$ such as the equation $\displaystyle y=\tan^{-1}(x)$ has a solution.