To determine if the function is continuous at $x = 2$, we need to check three conditions for continuity at this point:

1. The function must be defined at $x = 2$.
2. The left-hand limit as $x$ approaches 2 must equal the value of the function at $x = 2$.
3. The right-hand limit as $x$ approaches 2 must equal the value of the function at $x = 2$.

The given piecewise function is:

\begin{cases} x^2 + 1 & \text{if } x \leq 2 \\ 4x - 3 & \text{if } x > 2 \end{cases}$$ ### Step 1: Function at $$x = 2$$ Since the first piece applies when $$x \leq 2$$, we use the expression $$f(x) = x^2 + 1$$ to evaluate the function at $$x = 2$$: $$f(2) = 2^2 + 1 = 4 + 1 = 5$$ ### Step 2: Left-hand limit as $$x$$ approaches 2 The left-hand limit refers to the behavior of $$f(x)$$ as $$x$$ approaches 2 from values smaller than 2 (i.e., from the left side). For $$x \leq 2$$, the function is given by $$f(x) = x^2 + 1$$. Thus, we compute: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 + 1) = 2^2 + 1 = 5$$ ### Step 3: Right-hand limit as $$x$$ approaches 2 The right-hand limit refers to the behavior of $$f(x)$$ as $$x$$ approaches 2 from values larger than 2 (i.e., from the right side). For $$x > 2$$, the function is given by $$f(x) = 4x - 3$$. Thus, we compute: $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (4x - 3) = 4(2) - 3 = 8 - 3 = 5$$ ### Step 4: Conclusion We have: - $$f(2) = 5$$ - $$\lim_{x \to 2^-} f(x) = 5$$ - $$\lim_{x \to 2^+} f(x) = 5$$ Since the function is defined at $$x = 2$$, and both the left-hand and right-hand limits as $$x$$ approaches 2 are equal to the value of the function at $$x = 2$$, the function is **continuous** at $$x = 2$$. --- Would you like more details or have further questions? Here are 5 related questions for deeper exploration: 1. What happens if the function was defined differently at $$x = 2$$? 2. How can we prove a function is continuous over an interval? 3. What is the definition of continuity at a point in mathematical terms? 4. Can a function be continuous at some points and not at others? 5. How do piecewise functions impact the concept of continuity? **Tip**: When working with piecewise functions, always check the continuity conditions at the boundaries where the pieces meet!