which of the following accurately lists all discontinuities of the function below? f(x)={4, x< - 4; (x +…

which of the following accurately lists all discontinuities of the function below? f(x)={4, x< - 4; (x + 2)^2, - 4 <= x <= - 2; 1/2x + 1, - 2 < x < 4; - 1, x>4 point discontinuity at x = - 2 point discontinuity at x = 4; jump discontinuity at x = - 2 point discontinuities at x = - 4 and x = 4; jump discontinuity at x = - 2 jump discontinuities at x = - 4, x = - 2, and x = 4

which of the following accurately lists all discontinuities of the function below? f(x)={4, x< - 4; (x + 2)^2, - 4 <= x <= - 2; 1/2x + 1, - 2 < x < 4; - 1, x>4 point discontinuity at x = - 2 point discontinuity at x = 4; jump discontinuity at x = - 2 point discontinuities at x = - 4 and x = 4; jump discontinuity at x = - 2 jump discontinuities at x = - 4, x = - 2, and x = 4

Answer

Answer:

point discontinuities at (x=- 4) and (x = 4); jump discontinuity at (x=-2)

Explanation:

Step1: Check (x = - 4)

Left - hand limit: (\lim_{x\rightarrow - 4^{-}}f(x)=4), right - hand limit: (\lim_{x\rightarrow - 4^{+}}f(x)=( - 4 + 2)^{2}=4), but (f(-4)) is defined differently in the piece - wise function, so point discontinuity.

Step2: Check (x=-2)

Left - hand limit: (\lim_{x\rightarrow - 2^{-}}f(x)=( - 2 + 2)^{2}=0), right - hand limit: (\lim_{x\rightarrow - 2^{+}}f(x)=\frac{1}{2}( - 2)+1 = 0), but the function changes form, creating a jump.

Step3: Check (x = 4)

Left - hand limit: (\lim_{x\rightarrow 4^{-}}f(x)=\frac{1}{2}(4)+1=3), right - hand limit: (\lim_{x\rightarrow 4^{+}}f(x)=1), and (f(4)) has a different value in the piece - wise, so point discontinuity.