Math Problem Statement
The piecewise function f(x) is defined as:
f(x) =
x^2 - 9 for x ≤ 3,
(3 - x)^2 for x > 3.
Determine if the function is continuous at x = 3.
Solution
The image contains a piecewise function , which is defined as follows:
\begin{cases} x^2 - 9, & x \leq 3 \\ (3 - x)^2, & x > 3 \end{cases}$$ The text states that the function $$f(x)$$ is continuous at $$x = 3$$. ### To check the continuity of $$f(x)$$ at $$x = 3$$, we need to confirm the following conditions: 1. The left-hand limit as $$x$$ approaches 3 from the left, $$\lim_{x \to 3^-} f(x)$$. 2. The right-hand limit as $$x$$ approaches 3 from the right, $$\lim_{x \to 3^+} f(x)$$. 3. The value of the function at $$x = 3$$, $$f(3)$$. #### Step 1: Left-hand limit When $$x \leq 3$$, the function is defined as $$f(x) = x^2 - 9$$. So: $$\lim_{x \to 3^-} f(x) = 3^2 - 9 = 9 - 9 = 0.$$ #### Step 2: Right-hand limit When $$x > 3$$, the function is defined as $$f(x) = (3 - x)^2$$. So: $$\lim_{x \to 3^+} f(x) = (3 - 3)^2 = 0^2 = 0.$$ #### Step 3: Function value at $$x = 3$$ For $$x = 3$$, we use the first piece of the piecewise function, $$f(x) = x^2 - 9$$, since $$x \leq 3$$: $$f(3) = 3^2 - 9 = 9 - 9 = 0.$$ ### Conclusion: Since the left-hand limit, right-hand limit, and the function value at $$x = 3$$ all equal 0, the function $$f(x)$$ is indeed continuous at $$x = 3$$. --- Would you like more details or have any questions? Here are 5 related questions to consider: 1. How do you check if a function is differentiable at a point? 2. What are the conditions for a function to be continuous over a closed interval? 3. How would the result change if one of the limits did not match the function value at $$x = 3$$? 4. Can you extend the concept of continuity to a three-variable function? 5. How can we generalize the idea of continuity to piecewise functions in higher dimensions? **Tip**: Continuity at a point requires both the limits from the left and right to exist and be equal to the function value at that point.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
f(x) = x^2 - 9 for x ≤ 3
f(x) = (3 - x)^2 for x > 3
Left-hand limit: lim (x → 3^-) f(x)
Right-hand limit: lim (x → 3^+) f(x)
Continuity condition: Left-hand limit = Right-hand limit = f(3)
Theorems
Continuity Theorem
Limit Theorem
Suitable Grade Level
Grades 10-12