using the given graph of the function f, find the following. (a) the intercepts, if any (b) its domain and…

Explanation:

Step1: Find x - intercepts

The x - intercepts are the points where the graph crosses the x - axis. From the graph, the x - intercepts are $(-\pi,0)$ and $(\pi,0)$.

Step2: Find y - intercept

The y - intercept is the point where the graph crosses the y - axis. From the graph, the y - intercept is $(0,0)$.

Step3: Determine the domain

The domain is the set of all x - values for which the function is defined. From the graph, the domain is $[-\pi,\pi]$.

Step4: Determine the range

The range is the set of all y - values for which the function is defined. From the graph, the range is $[-6,6]$.

Step5: Find intervals of increase, decrease, and constancy

The function is increasing on the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, decreasing on the intervals $\left[-\pi,-\frac{\pi}{2}\right]$ and $\left[\frac{\pi}{2},\pi\right]$, and there are no intervals where it is constant.

Step6: Check if the function is even, odd, or neither

A function $y = f(x)$ is even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$. Since the graph is symmetric about the origin (if we rotate the graph 180 degrees about the origin, it looks the same), the function is odd.

Answer:

(a) $(-\pi,0),(0,0),(\pi,0)$ (b) $[-\pi,\pi]$ (c) Increasing: $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, Decreasing: $\left[-\pi,-\frac{\pi}{2}\right],\left[\frac{\pi}{2},\pi\right]$, Constant: None (d) Odd