(a) graph the given function, (b) find all values of x where the function is discontinuous, and (c) find the…

Explanation:

Step1: Analyze each - part of the piece - wise function for graphing

For (g(x)= - 2) when (x\lt - 3), it is a horizontal line (y = - 2) for (x) values less than (-3). For (g(x)=x^{2}-3) when (-3\leq x\leq1), it is a parabola (y = x^{2}-3) with vertex at ((0, - 3)) and we consider the part of the parabola for (x) in the interval ([-3,1]). For (g(x)= - 2) when (x\gt1), it is a horizontal line (y = - 2) for (x) values greater than (1).

Step2: Check continuity at (x=-3)

Left - hand limit as (x\to - 3^{-}): (\lim_{x\to - 3^{-}}g(x)=-2). Right - hand limit as (x\to - 3^{+}): (\lim_{x\to - 3^{+}}g(x)=(-3)^{2}-3=9 - 3 = 6). Since (\lim_{x\to - 3^{-}}g(x)\neq\lim_{x\to - 3^{+}}g(x)), the function is discontinuous at (x=-3).

Step3: Check continuity at (x = 1)

Left - hand limit as (x\to1^{-}): (\lim_{x\to1^{-}}g(x)=1^{2}-3=-2). Right - hand limit as (x\to1^{+}): (\lim_{x\to1^{+}}g(x)=-2). Since (\lim_{x\to1^{-}}g(x)=\lim_{x\to1^{+}}g(x)=-2), the function is continuous at (x = 1).

Step4: Find left - hand and right - hand limits at (x=-3)

Left - hand limit as (x\to - 3^{-}): (\lim_{x\to - 3^{-}}g(x)=-2). Right - hand limit as (x\to - 3^{+}): (\lim_{x\to - 3^{+}}g(x)=(-3)^{2}-3 = 6).

(a) To graph: The graph has a horizontal line (y=-2) for (x\lt - 3), a parabola (y = x^{2}-3) for (-3\leq x\leq1) (vertex at ((0,-3))), and a horizontal line (y = - 2) for (x\gt1). (b) The function (g(x)) is discontinuous at (x=-3). (c) At (x=-3), (\lim_{x\to - 3^{-}}g(x)=-2) and (\lim_{x\to - 3^{+}}g(x)=6).

Since there are no graph options provided in a proper way to choose from, we focus on the non - graph parts:

Answer:

(b) (x=-3) (c) (\lim_{x\to - 3^{-}}g(x)=-2), (\lim_{x\to - 3^{+}}g(x)=6)