determine whether the graph is that of a function by using the vertical - line test. in either case, use the…

Explanation:

Step1: Check if it's a function

The vertical - line test states that if any vertical line intersects the graph at more than one point, it is not a function. A vertical line $x = 4$ intersects the given graph at two points, so it is not a function.

Step2: Find the domain

The domain is the set of all $x$ - values of the graph. Looking at the graph, the $x$ - values start from $x = 2$ and go to positive infinity. So, the domain is $[2,\infty)$.

Step3: Find the range

The range is the set of all $y$ - values of the graph. Since the graph extends infinitely in both the positive and negative $y$ - directions, the range is $(-\infty,\infty)$.

Step4: Find the intercepts

The $x$ - intercept is the point where the graph crosses the $x$ - axis ($y = 0$) and the $y$ - intercept is the point where the graph crosses the $y$ - axis ($x = 0$). From the graph, the $x$ - intercept is $(2,0)$ and there is no $y$ - intercept as the graph does not cross the $y$ - axis.

Step5: Check for symmetry

• For $x$ - axis symmetry: If $(x,y)$ is on the graph, then $(x, - y)$ should be on the graph. The graph does not have $x$ - axis symmetry.
• For $y$ - axis symmetry: If $(x,y)$ is on the graph, then $(-x,y)$ should be on the graph. The graph does not have $y$ - axis symmetry since the graph starts at $x = 2$ and has no part for negative $x$ values.
• For origin symmetry: If $(x,y)$ is on the graph, then $(-x,-y)$ should be on the graph. The graph does not have origin symmetry.

Answer:

(a) Domain: $[2,\infty)$; Range: $(-\infty,\infty)$ (b) The intercept(s) is/are $(2,0)$ (c) There is no symmetry with respect to the $x$ - axis, $y$ - axis, or the origin.