1-96. graph each piecewise - defined function given below and determine whether the function is…

Explanation:

Step1: Analyze left - hand limit for part a

For (x\to - 2^{-}), (f(x)=x + 4). So (\lim_{x\to - 2^{-}}f(x)=\lim_{x\to - 2^{-}}(x + 4)=-2 + 4=2).

Step2: Analyze right - hand limit for part a

For (x\to - 2^{+}), (f(x)=\frac{1}{2}x^{2}). So (\lim_{x\to - 2^{+}}f(x)=\frac{1}{2}\times(-2)^{2}=2). And (f(-2)=\frac{1}{2}\times(-2)^{2}=2). Since (\lim_{x\to - 2^{-}}f(x)=\lim_{x\to - 2^{+}}f(x)=f(-2)), the function is continuous at (x=-2).

Step3: Analyze left - hand limit for part b

For (x\to3^{-}), (f(x)=\frac{1}{2}x^{2}). So (\lim_{x\to3^{-}}f(x)=\frac{1}{2}\times3^{2}=\frac{9}{2}).

Step4: Analyze right - hand limit for part b

For (x\to3^{+}), (f(x)=x + 4). So (\lim_{x\to3^{+}}f(x)=3 + 4=7). Since (\lim_{x\to3^{-}}f(x)\neq\lim_{x\to3^{+}}f(x)), the function is discontinuous at (x = 3).

Answer:

a. The function is continuous. b. The function is discontinuous.