1. **State the problem:** We have a piecewise function defined as $$f(x) = \begin{cases} 3x + 1 & \text{if } x < -2 \\ x - 3 & \text{if } x \geq -2 \end{cases}$$ We need to graph this function and determine if it is continuous. 2. **Graph the function:** - For $x < -2$, the function is $f(x) = 3x + 1$. This is a line with slope 3 and y-intercept 1. - For $x \geq -2$, the function is $f(x) = x - 3$. This is a line with slope 1 and y-intercept $-3$. 3. **Evaluate the function at the boundary $x = -2$:** - Left-hand limit: $\lim_{x \to -2^-} f(x) = 3(-2) + 1 = -6 + 1 = -5$ - Right-hand limit and value: $f(-2) = (-2) - 3 = -5$ 4. **Check continuity:** A function is continuous at $x = -2$ if $$\lim_{x \to -2^-} f(x) = f(-2) = \lim_{x \to -2^+} f(x)$$ Since both limits and the function value equal $-5$, the function is continuous at $x = -2$. 5. **Summary:** - The graph consists of two line segments meeting at $(-2, -5)$. - The left segment has an open circle at $(-2, -5)$ because the definition is for $x < -2$. - The right segment has a closed circle at $(-2, -5)$ because the definition includes $x = -2$. - The function is continuous at $x = -2$. **Final answer:** The function is continuous.