Math Problem Statement
Solution
To ensure the function is continuous across the entire real number line, we need to make sure it is continuous at . This requires the left-hand limit and the right-hand limit at to be equal, as well as equal to the function value at that point.
The function is given by:
\begin{cases} x^2, & \text{if } x \leq 2 \\ ax + 2, & \text{if } x > 2 \end{cases}$$ ### Step 1: Continuity Condition at $$x = 2$$ To ensure continuity, the following must hold: $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2).$$ **Left-hand limit:** $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4.$$ **Right-hand limit:** $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (ax + 2) = 2a + 2.$$ **Function value at $$x = 2$$:** $$f(2) = 2^2 = 4.$$ ### Step 2: Equating the Limits For continuity at $$x = 2$$, we set: $$2a + 2 = 4.$$ ### Step 3: Solve for $$a$$ $$2a = 4 - 2,$$ $$2a = 2,$$ $$a = 1.$$ Thus, the function will be continuous on the entire real number line if $$a = 1$$. Would you like a more detailed explanation, or do you have any other questions? Here are some follow-up questions you might find interesting: 1. What would happen if $$a$$ were a different value? 2. How do we determine continuity for piecewise functions in general? 3. Can the function $$f(x)$$ be differentiable at $$x = 2$$ when $$a = 1$$? 4. How would the graph of $$f(x)$$ look around $$x = 2$$? 5. What are the implications of a function being continuous on the real line? **Tip:** When dealing with piecewise functions, always check the boundaries where the definition changes, as continuity and differentiability issues often arise there.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Piecewise Functions
Formulas
Left-hand limit: lim_{x→2^-} f(x)
Right-hand limit: lim_{x→2^+} f(x)
Continuity condition: lim_{x→c} f(x) = f(c)
Theorems
Continuity Theorem
Suitable Grade Level
Grades 11-12
Related Recommendation
Find the Value of a for Continuity in a Piecewise Function
Find Parameter a for Continuity of a Piecewise Function
Find the Value of 'a' for a Continuous Function at x = -2
Determining the Constant 'a' for Continuity of a Piecewise Function
Determine Continuity for a Piecewise Function: Solve for Constant a