Chapter 4.7, Problem 35E

Solution

Analysing function for domain, range, continuity, increasing, decreasing.

• Domain: $(-\infty ,\infty )$
• Range: $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
• Continuous
• Increasing
• Symmetric with respect to origin (odd function).
• Bounded above and below.
• No local extremes.
• Horizontal asymptotes: $y=-\frac{\pi }{2}$ and $y=\frac{\pi }{2}$, no vertical asymptotes
• $\underset{x\to -\infty }{\lim }{\tan }^{-1}x=-\frac{\pi }{2}$ and $\underset{x\to \infty }{\lim }{\tan }^{-1}x=\frac{\pi }{2}$

Given information: function $f(x)={\tan }^{-1}x$ .

Calculation: Obtain a graph of $f(x)={\tan }^{-1}x$ as shown:

  

The $x$ -values extend to the left and to the right so the domain is all real numbers: $(-\infty ,\infty )$

The $y$ -values have an interval from $-\frac{\pi }{2}$ to $-\frac{\pi }{2}$ , inclusive, so range is:

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

The graph is continuous on its domain (no gaps or breaks) and is increasing. The graph is symmetric with respect to origin so it is an odd function. The graph is bounded above as well as below.

There is no local extreme (maximum or minimum).

There are horizontal asymptotes at $y=-\frac{\pi }{2}$ and $y=\frac{\pi }{2}$ .

There are no vertical asymptotes.

End behaviour: $\underset{x\to -\infty }{\lim }{\tan }^{-1}x=-\frac{\pi }{2}$ and $\underset{x\to \infty }{\lim }{\tan }^{-1}x=\frac{\pi }{2}$

• Domain: $(-\infty ,\infty )$
• Range: $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
• Continuous
• Increasing
• Symmetric with respect to origin (odd function).
• Bounded above and below.
• No local extremes.
• Horizontal asymptotes: $y=-\frac{\pi }{2}$ and $y=\frac{\pi }{2}$, no vertical asymptotes.
• $\underset{x\to -\infty }{\lim }{\tan }^{-1}x=-\frac{\pi }{2}$ and $\underset{x\to \infty }{\lim }{\tan }^{-1}x=\frac{\pi }{2}$