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 $x =-\pi,0,\pi$. In ordered - pair form, they are $(-\pi,0),(0,0),(\pi,0)$.

Step2: Find y - intercept

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

Step3: Determine the domain

The domain is the set of all possible x - values. From the graph, the x - values range from $-\pi$ to $\pi$. In interval notation, the domain is $[-\pi,\pi]$.

Step4: Determine the range

The range is the set of all possible y - values. The minimum y - value is $y=-3$ and the maximum y - value is $y = 3$. In interval notation, the range is $[-3,3]$.

Step5: Find intervals of increase, decrease, and constancy

The function is increasing on the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and decreasing on the intervals $\left(-\pi,-\frac{\pi}{2}\right)$ and $\left(\frac{\pi}{2},\pi\right)$. There are no intervals where the function 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) Domain: $[-\pi,\pi]$, Range: $[-3,3]$ (c) Increasing on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, Decreasing on $\left(-\pi,-\frac{\pi}{2}\right)$ and $\left(\frac{\pi}{2},\pi\right)$ (d) Odd