Chapter 5, Problem 11RCC

a.

To determine

To find: The definition, domain and range of the inverse tangent function.

Explanation of Solution

Given information:

The given function is

  ${\tan }^{-1}x$

Definition of inverse tangent function: The inverse tangent function $\left({\tan }^{-1}\right)$ can be used to calculate the angle by providing the ratio of the two sides of the triangle. The inverse of the tangent function lies in the 1^st and 4^th quadrants.

Domain: $\left(-\infty ,\infty \right)$

Range: $\left(\frac{-\pi }{2},\frac{\pi }{2}\right)$

b.

To determine

To find:the value of $x$ .

Answer to Problem 11RCC

The value of the $x$ is vary from $-\infty$ to $\infty$ .

Explanation of Solution

Given information:

  $\tan \left({\tan }^{-1}x\right)=x$

Consider the given function,

  $\tan \left({\tan }^{-1}x\right)=x$

The above expression is true for domain of the tangent inverse function. Since, the domain of the tangent inverse function varies from $-\infty$ to $\infty$ . This means the above function is valid for all real number.

c.

To determine

To find:the value of $x$ .

Answer to Problem 11RCC

The given function is valid for $\left(\frac{-\pi }{2},\frac{\pi }{2}\right)$ .

Explanation of Solution

Given information:

  ${\tan }^{-1}\left(\tan x\right)=x$

Consider the given function,

  ${\tan }^{-1}\left(\tan x\right)=x$

The minimum value of the $\tan x$ is $-\infty$ , for this the value of the ${\tan }^{-1}$ is $\frac{-\pi }{2}$ .The maximum value of the $\tan x$ is $\infty$ , for this the value of the ${\tan }^{-1}$ is $\frac{\pi }{2}$ . Thus, the above function is valid for $\left(\frac{-\pi }{2},\frac{\pi }{2}\right)$ .