sketch by hand the graph of the function. determine if f is continuous. f(x)={36 if x < -6; x² if -6 ≤ x ≤…

sketch by hand the graph of the function. determine if f is continuous. f(x)={36 if x < -6; x² if -6 ≤ x ≤ 6; 42 - x if x > 6} is f continuous? yes no choose the correct graph of the function below.

sketch by hand the graph of the function. determine if f is continuous. f(x)={36 if x < -6; x² if -6 ≤ x ≤ 6; 42 - x if x > 6} is f continuous? yes no choose the correct graph of the function below.

Answer

Explanation:

Step1: Analyze left - hand limit at x = - 6

For (x\lt - 6), (f(x)=36). As (x) approaches (-6) from the left ((x\to - 6^{-})), (\lim_{x\to - 6^{-}}f(x)=36). For (-6\leq x\leq6), (f(x)=x^{2}), and (\lim_{x\to - 6^{+}}f(x)=(-6)^{2}=36).

Step2: Analyze left - hand limit at x = 6

For (-6\leq x\leq6), (f(x)=x^{2}), so (\lim_{x\to6^{-}}f(x)=6^{2}=36). For (x > 6), (f(x)=42 - x), and (\lim_{x\to6^{+}}f(x)=42-6 = 36). Also, (f(-6)=(-6)^{2}=36) and (f(6)=6^{2}=36).

Step3: Check continuity conditions

Since (\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=f(a)) for (a=-6) and (a = 6), and the function is well - defined and continuous on each of the open intervals ((-\infty,-6)), ((-6,6)) and ((6,\infty)), the function (f(x)) is continuous.

Answer:

Yes