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 value of y is $-4$ and the maximum value of y is $4$. In interval notation, the range is $[-4,4]$.

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 $f(-x)=f(x)$ (symmetric about the y - axis) and odd if $f(-x)=-f(x)$ (symmetric about the origin). The graph is symmetric about the origin. For example, $f\left(-\frac{\pi}{2}\right)=-4$ and $f\left(\frac{\pi}{2}\right)=4$. So the function is odd.

Answer:

(a) $(-\pi,0),(0,0),(\pi,0)$ (b) Domain: $[-\pi,\pi]$, Range: $[-4,4]$ (c) Increasing: $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, Decreasing: $\left[-\pi,-\frac{\pi}{2}\right]\cup\left[\frac{\pi}{2},\pi\right]$, Constant: None (d) Odd