Chapter 4.7, Problem 34E

Solution

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

Domain: $[-1.1]$

Range: $\left[0.\pi \right]$

• Continuous
• Increasing
• The function is symmetric with respect to the Origin (odd function).
• Bounded above and below.
• The absolute maximum is at $(-1,\pi )$ and The absolute minimum is at $(1,0)$ .
• No vertical and horizontal asymptotes.
• No end behaviour.

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

Calculation: Obtain a graph of $f(x)={\cos }^{-1}x$ 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 $0$ to $\pi$ , inclusive, so range is:

  $\left[0.\pi \right]$

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 an absolute maximum of $\pi$ when $x=1$ and an absolute minimum of $0$ 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: $\left[0.\pi \right]$

• Continuous
• Increasing
• The function is symmetric with respect to the Origin (odd function).
• Bounded above and below.
• The absolute maximum is at $(-1,\pi )$ and the absolute minimum is at $(1,0)$ .
• No vertical and horizontal asymptotes.
• No end behaviour.