Answer
$\text{(a)}$ domain: $(-\infty,4]$; range: $(-\infty,3]$
$\text{(b)}$ increasing at $(-\infty,-2)\cup(2,4)$; decreasing at $(-2,2)$
$\text{(c)}$ local minimum $=-1$; local maximum $=1$.
$\text{(d)}$ absolute maximum $=3$
$\text{(e)}$ no symmetry
$\text{(f)}$ $\text{neither odd or even}$
$\text{(g)}$ $x$-intercepts: $-3, 0, 3$; $y$-intercept: $0$
Work Step by Step
$\text{(a)}\quad$ The domain of a function is the set of all $x$ values while the range of a function is the set of all $y$ values.
The graph shows that the function includes the $x$ values from $4$ and below.
Thus, the domain is $(-\infty,4]$.
The graph also shows that the function includes the $y$ values from $3$ and below.
Thus, the range is $(-\infty,3]$.
$\text{(b)}\quad$ The graph shows that the function is increasing in the intervals $(-\infty,-2)\cup(2,4)$, and decreasing in the interval $(-2,2)$. The function has no interval where its value is a contant.
$\text{(c)}\quad$ The function has a local minimum of $-1$ at $x=2$, and has a local maximum of $1$ at $x=-2$.
$\text{(d)}\quad$ The function has an absolute maximum of $3$ at $x=4$ but it has no absolute minimum.
$\text{(e)}\quad$ A graph is symmetric with respect to the $x$-axis if for every point $(x, y)$ on the graph, the point $(x, -y)$ is also on the graph. Note that $(-4, -3)$ is on the graph but $(-4, 3)$ is not. Hence, the graph is not symmetric with respect to the $x$-axis.
A graph is symmetric with respect to the $y$-axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph. Note that $(-4, -3)$ is on the graph but $(4, -3)$ is not. Hence, the graph is not symmetric with respect to the $y$-axis.
A graph is symmetric with respect to the origin if for every point $(x, y)$ on the graph, the point $(-x, -y)$ is also on the graph. Note that for all the points from $x=-4$ to $x=4$, the graph is symmetric with respect to the origin. However, for points where $x<-4$, they no longer have a corresponding symmetric point in the interval $x>4$. Hence, the graph as a whole is not symmetric with respect to the origin.
$\text{(f)}\quad$ A function is odd if it is symmetric with respect to the origin, while a function is even if its graph is symmetric with respect to the $$y$-axis. Since the function has no symmetry, then it is neither odd nor even.
$\text{(g)}\quad$ The $x$-intercept/s are the $x$-coordinates of the points where the value of $y$ is $0$. Thus, the $x$-intercepts are $-3, 0, \text{ and } 3$.
The $y$-intercept is the $y$-value of the point where $x=0$. Thus, the $y$-intercept is $0$.
