sketch by hand the graph of the function. determine if f is continuous.\n f(x)=\begin{cases}36 & \text{if }…

Explanation:

Step1: Analyze the first - part of the function

For (x < - 6), (f(x)=36), which is a horizontal line (y = 36) for all (x) values less than (-6).

Step2: Analyze the second - part of the function

For (-6\leq x\leq6), (f(x)=x^{2}). This is a parabola with vertex at ((0,0)). When (x=-6), (y = (-6)^{2}=36); when (x = 6), (y=6^{2}=36).

Step3: Analyze the third - part of the function

For (x>6), (f(x)=42 - x). When (x = 6), (f(6)=36) and as (x) increases, the function is a line with slope (-1). For example, when (x=7), (f(7)=42 - 7=35).

Step4: Check for continuity

The left - hand limit as (x\to - 6) is (36), (f(-6)=(-6)^{2}=36), so it is continuous at (x=-6). The left - hand limit as (x\to6) is (6^{2}=36), and the right - hand limit as (x\to6) is (42-6 = 36), and (f(6)=36), so the function is continuous.

Answer:

Based on the above - mentioned analysis of the function's behavior in different intervals and continuity, we need to visually match the graph. Without seeing the actual graphs A, B, C, D in detail, we know that the graph should have a horizontal line (y = 36) for (x < - 6), a parabola (y=x^{2}) for (-6\leq x\leq6) and a line (y = 42 - x) for (x>6) all connected smoothly. You need to choose the graph that shows these characteristics.