To determine if the function $f(x)$ is continuous, we need to check the continuity at the points where the pieces of the function connect, which are $x = -3$ and $x = 2$. A function is continuous if the left-hand limit, right-hand limit, and the value of the function are all equal at the points of interest.

The function is defined as:

\begin{cases} -2x & \text{if } x < -3 \\ 3x - 1 & \text{if } -3 \leq x \leq 2 \\ -4x & \text{if } x > 2 \end{cases}$$ ### 1. Check continuity at $$x = -3$$ - **Left-hand limit (as $$x \to -3^-$$):** $$\lim_{x \to -3^-} f(x) = \lim_{x \to -3^-} -2x = -2(-3) = 6$$ - **Right-hand limit (as $$x \to -3^+$$):** $$\lim_{x \to -3^+} f(x) = \lim_{x \to -3^+} (3x - 1) = 3(-3) - 1 = -9 - 1 = -10$$ - **Value of the function at $$x = -3$$:** $$f(-3) = 3(-3) - 1 = -10$$ Since the left-hand limit ($$6$$) does not equal the right-hand limit ($$-10$$), the function is **not continuous at $$x = -3$$**. --- ### 2. Check continuity at $$x = 2$$ - **Left-hand limit (as $$x \to 2^-$$):** $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (3x - 1) = 3(2) - 1 = 6 - 1 = 5$$ - **Right-hand limit (as $$x \to 2^+$$):** $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} -4x = -4(2) = -8$$ - **Value of the function at $$x = 2$$:** $$f(2) = 3(2) - 1 = 5$$ Since the left-hand limit ($$5$$) does not equal the right-hand limit ($$-8$$), the function is **not continuous at $$x = 2$$**. --- ### Conclusion: The function $$f(x)$$ is **not continuous** because it fails the continuity condition at both $$x = -3$$ and $$x = 2$$. Would you like further clarification or a graphical representation of this function? --- #### 5 Related Questions: 1. What are the conditions for a function to be continuous at a point? 2. How can we analyze the continuity of piecewise functions in general? 3. What role do limits play in determining continuity? 4. Could this function be made continuous by modifying it at $$x = -3$$ and $$x = 2$$? 5. How does differentiability relate to continuity for piecewise functions? #### Tip: For piecewise functions, always pay close attention to the points where the pieces connect—these are the most likely places for discontinuities!