Chapter 4.7, Problem 33E

Solution

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

Domain: $[-1.1]$

Range: $\left[-\frac{\pi }{2}.\frac{\pi }{2}\right]$

• Continuous
• Increasing
• Symmetric with respect to the origin (odd function).
• Bounded above and below.
• Absolute maximum at $\left(1,\frac{\pi }{2}\right)$ and absolute minimum at $\left(-1,-\frac{\pi }{2}\right)$ .
• No vertical and horizontal asymptotes.
• No end behaviour.

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

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

  

The $x$ -values have an internal from $-1$ to $1$ , inclusive, so the domain is $[-1.1]$

The $y$ -values have an internal 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 the origin so it is an odd function. The graph is bounded above and below.

There is an absolute maximum of $\frac{\pi }{2}$ when $x=1$ and an absolute minimum of $-\frac{\pi }{2}$ ,When $x=-1$ .

There are no asymptotes, since the domain is bounded, there is no end behaviour.

Hence, the overall values are:

Domain: $[-1.1]$

Range: $\left[-\frac{\pi }{2}.\frac{\pi }{2}\right]$

• Continuous
• Increasing
• Symmetric with respect to the origin (odd function).
• Bounded above and below.
• Absolute maximum at $\left(1,\frac{\pi }{2}\right)$ and absolute minimum at $\left(-1,-\frac{\pi }{2}\right)$ .
• No vertical and horizontal asymptotes.
• No end behaviour.