1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} x + 1 & \text{if } x < 4 \\ (x - 4)^2 + 3 & \text{if } x \geq 4 \end{cases}$$ We want to analyze this function, typically checking continuity and behavior at the breakpoint $x=4$. 2. **Check continuity at $x=4$:** - The left-hand limit as $x$ approaches 4 is $\lim_{x \to 4^-} f(x) = 4 + 1 = 5$. - The right-hand limit as $x$ approaches 4 is $\lim_{x \to 4^+} f(x) = (4 - 4)^2 + 3 = 0 + 3 = 3$. - The function value at $x=4$ is $f(4) = 3$. 3. **Interpretation:** Since the left-hand limit (5) does not equal the right-hand limit and the function value (3), the function is **not continuous** at $x=4$. 4. **Graph shape:** - For $x<4$, the graph is a straight line with slope 1 and y-intercept 1. - For $x \geq 4$, the graph is a parabola opening upwards with vertex at $(4,3)$. 5. **Summary:** The function has a jump discontinuity at $x=4$ because the two pieces do not meet at the same value. **Final answer:** The function $f(x)$ is not continuous at $x=4$ because $\lim_{x \to 4^-} f(x) \neq \lim_{x \to 4^+} f(x)$.