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
(type your answer in interval notation.)
(b) select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the intercept(s) is/are (-5,0),(5,0). (type an ordered pair. use a comma to separate answers as needed.)
b. there are no intercepts.
(c) select all that apply.
a. the graph is symmetrical with respect to the origin.
b. the graph is symmetrical with respect to the x - axis.
c. the graph is symmetrical with respect to the y - axis.
d. the graph is not symmetrical.

Explanation:

Step1: Check if it's a function

Use vertical - line test. If any vertical line intersects the graph at more than one point, it's not a function. From the graph, a vertical line can intersect at multiple points, so it's not a function.

Step2: Find domain

The set of all x - values of the graph. Looking at the graph, the x - values range from negative infinity to positive infinity. Domain: $(-\infty,\infty)$

Step3: Find range

The set of all y - values of the graph. The y - values range from negative infinity to positive infinity. Range: $(-\infty,\infty)$

Step4: Identify intercepts

The x - intercepts are the points where the graph crosses the x - axis. From the given choice, the x - intercepts are $(-5,0)$ and $(5,0)$.

Step5: Check symmetry

For x - axis symmetry, if $(x,y)$ is on the graph, then $(x, - y)$ is on the graph. For y - axis symmetry, if $(x,y)$ is on the graph, then $(-x,y)$ is on the graph. For origin symmetry, if $(x,y)$ is on the graph, then $(-x,-y)$ is on the graph. The graph is symmetric about the x - axis as for every point $(x,y)$ there is a point $(x, - y)$.

Answer:

(a) Domain: $(-\infty,\infty)$; Range: $(-\infty,\infty)$
(b) A. The intercept(s) is/are $(-5,0),(5,0)$
(c) B. The graph is symmetrical with respect to the x - axis.