To determine the discontinuities of the given piecewise function, we need to evaluate the continuity at the transition points, specifically at [tex]\( x = -4 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 4 \)[/tex].
Let's consider each of these points one by one:
1. At [tex]\( x = -4 \)[/tex]:
- For [tex]\( x < -4 \)[/tex]: [tex]\( f(x) = 4 \)[/tex].
- For [tex]\( -4 \leq x \leq -2 \)[/tex]: [tex]\( f(x) = (x + 2)^2 \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\(-4\)[/tex] from the left, [tex]\( f(x) \)[/tex] approaches 4.
- As [tex]\( x \)[/tex] approaches [tex]\(-4\)[/tex] from the right, [tex]\( f(x) = (-4 + 2)^2 = 4 \)[/tex].
Since both the left-hand limit and right-hand limit are equal and [tex]\( f(-4) = 4 \)[/tex], the function is continuous at [tex]\( x = -4 \)[/tex].
2. At [tex]\( x = -2 \)[/tex]:
- For [tex]\( -4 \leq x \leq -2 \)[/tex]: [tex]\( f(x) = (x + 2)^2 \)[/tex].
- For [tex]\( -2 < x < 4 \)[/tex]: [tex]\( f(x) = -\frac{1}{2} x + 1 \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the left, [tex]\( f(x) = (-2 + 2)^2 = 0 \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the right, [tex]\( f(x) = -\frac{1}{2} \cdot (-2) + 1 = 2 \)[/tex].
Since the left-hand limit (0) is not equal to the right-hand limit (2) as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex], there is a jump discontinuity at [tex]\( x = -2 \)[/tex].
3. At [tex]\( x = 4 \)[/tex]:
- For [tex]\( -2 < x < 4 \)[/tex]: [tex]\( f(x) = -\frac{1}{2} x + 1 \)[/tex].
- For [tex]\( x > 4 \)[/tex]: [tex]\( f(x) = -1 \)[/tex].
- As [tex]\( x \)[/tex] approaches \4 from the left, [tex]\( f(x) = -\frac{1}{2} \cdot 4 + 1 = -1 \)[/tex].
- As [tex]\( x \)[/tex] approaches \4 from the right, [tex]\( f(x) = -1 \)[/tex].
Since both the left-hand limit and right-hand limit are equal and [tex]\( f(4) = -1 \)[/tex], the function is continuous at [tex]\( x = 4 \)[/tex].
Given this analysis, the function [tex]\( f(x) \)[/tex] has a jump discontinuity at [tex]\( x = -2 \)[/tex].
Therefore, the accurate listing of all discontinuities of the function is:
- Jump discontinuity at [tex]\( x = -2 \)[/tex]
Hence, the correct option is:
- jump discontinuity at [tex]\( x = -2 \)[/tex].
