Questions: Determine the (x) values where the following function is discontinuous (f(x)=leftbeginarraylrr -2 x+6 text if x leq-4 -3 x+2 text if -4<x leq 1 2 x-3 text if 1<x endarrayright.)

Solution Steps

To determine the $x$ values where the function is discontinuous, we need to check the points where the pieces of the function meet, specifically at $x = -4$ and $x = 1$. We will evaluate the left-hand limit, right-hand limit, and the function value at these points to see if they match.

To determine where the function \( f(x) \) is discontinuous, we need to check the points where the pieces of the function meet. Specifically, these points are \( x = -4 \) and \( x = 1 \).

• Left-hand limit as \( x \) approaches \(-4\): \[ \lim_{{x \to -4^-}} f(x) = -2(-4) + 6 = 8 + 6 = 14 \]
• Right-hand limit as \( x \) approaches \(-4\): \[ \lim_{{x \to -4^+}} f(x) = -3(-4) + 2 = 12 + 2 = 14 \]
• Function value at \( x = -4 \): \[ f(-4) = -2(-4) + 6 = 8 + 6 = 14 \]

Since the left-hand limit, right-hand limit, and function value at \( x = -4 \) are all equal, the function is continuous at \( x = -4 \).

• Left-hand limit as \( x \) approaches \( 1 \): \[ \lim_{{x \to 1^-}} f(x) = -3(1) + 2 = -3 + 2 = -1 \]
• Right-hand limit as \( x \) approaches \( 1 \): \[ \lim_{{x \to 1^+}} f(x) = 2(1) - 3 = 2 - 3 = -1 \]
• Function value at \( x = 1 \): \[ f(1) = -3(1) + 2 = -3 + 2 = -1 \]

Since the left-hand limit, right-hand limit, and function value at \( x = 1 \) are all equal, the function is continuous at \( x = 1 \).

Final Answer