Chapter 7.1, Problem 12AYU

To determine

To define: Whether the statement “ $y={\tan }^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\frac{\pi }{2}<y<\frac{\pi }{2}$ ” is true or false.

Answer to Problem 12AYU

The statement “ $y={\tan }^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\frac{\pi }{2}<y<\frac{\pi }{2}$ ” is true.

Explanation of Solution

Given information:

The statement “ $y={\tan }^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .”

Consider the provided statement “ $y={\tan }^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .”

Recall thatin the function $y={\tan }^{-1}x$ , $y$ is the angle whose tangent is $x$ .

The implicit form of the inverse function is $x=\tan y$ .

The domain of the function is the set of input function and range is the set of output function.

The domain of the function $y={\tan }^{-1}x$ is $-\infty <x<\infty$ and it’s range is $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Thus, the statement “ $y={\tan }^{-1}x$ means $x=\tan y$ , where $-\infty <x<\infty$ and $-\frac{\pi }{2}<y<\frac{\pi }{2}$ ” is true.