Question

determine whether the graph is that of a function by using the vertical - line test. in either case, use the graph to find the following. (a) the domain and range (b) the intercepts, if any. (c) any symmetry with respect to the x - axis, y - axis, or the origin is the graph that of a function? a. yes, the graph is a function because every vertical line intersects the graph in at most one point. b. no, the graph is not a function because a vertical line x = 2 intersects the graph at only one point. c. no, the graph is not a function because a vertical line x = 2 intersects the graph at two points. d. yes, the graph is a function because every vertical line intersects the graph in more than one point. (a) the domain is . (type your answer in interval notation.)

Explanation:

Step1: Check function using vertical - line test

By the vertical - line test, if any vertical line intersects the graph at most once, it's a function. Here, every vertical line intersects the graph in at most one point, so it's a function.

Step2: Find domain

The x - values of the graph range from - 4 to 4. So the domain is $[-4,4]$.

Step3: Find x - intercepts

Set $y = 0$. The graph intersects the x - axis at $x=-4$ and $x = 4$. So the x - intercepts are $(-4,0)$ and $(4,0)$.

Step4: Find y - intercepts

Set $x = 0$. The graph intersects the y - axis at $y = 8$. So the y - intercept is $(0,8)$.

Step5: Check symmetry

Replace $x$ with $-x$. The equation of the graph is symmetric about the y - axis since for every point $(x,y)$ on the graph, $(-x,y)$ is also on the graph. It is not symmetric about the x - axis (since replacing $y$ with $-y$ changes the graph) and not symmetric about the origin (since replacing $x$ with $-x$ and $y$ with $-y$ changes the graph).

Answer:

A. Yes, the graph is a function because every vertical line intersects the graph in at most one point.
(a) $[-4,4]$
(b) x - intercepts: $(-4,0),(4,0)$; y - intercept: $(0,8)$
(c) Symmetric about the y - axis.