Math Problem Statement
Tentukan a yang memenuhi agar fungsi di bawah ini kontinu: f(x) = {3x + 4, x ≤ a; x + 3, x > a}
Solution
The problem in the image asks to determine the value of such that the function is continuous. The function is defined as:
3x + 4, & x \leq a \\ x + 3, & x > a \end{cases}$$ To ensure the function is continuous at $$x = a$$, the left-hand limit (as $$x \to a^-$$) must equal the right-hand limit (as $$x \to a^+$$), and both must equal $$f(a)$$. ### Step 1: Left-hand limit ($$x \leq a$$) For $$x \leq a$$, the function is $$f(x) = 3x + 4$$. So the limit as $$x \to a^-$$ is: $$\lim_{x \to a^-} f(x) = 3a + 4$$ ### Step 2: Right-hand limit ($$x > a$$) For $$x > a$$, the function is $$f(x) = x + 3$$. So the limit as $$x \to a^+$$ is: $$\lim_{x \to a^+} f(x) = a + 3$$ ### Step 3: Continuity condition For $$f(x)$$ to be continuous at $$x = a$$, the two limits must be equal: $$3a + 4 = a + 3$$ ### Step 4: Solve for $$a$$ Now, solve this equation for $$a$$: $$3a + 4 = a + 3$$ $$3a - a = 3 - 4$$ $$2a = -1$$ $$a = -\frac{1}{2}$$ Thus, the value of $$a$$ that makes the function continuous is $$a = -\frac{1}{2}$$. Would you like more details or have any further questions? Here are 5 related questions to expand your understanding: 1. What does it mean for a function to be continuous at a point? 2. How do you check continuity for piecewise functions at a point? 3. What happens if the function isn't continuous at the point where the rule changes? 4. Can you explain the difference between one-sided limits and two-sided limits? 5. What are some practical applications of continuous functions in real-world scenarios? **Tip**: Continuity of a function ensures there are no sudden jumps or breaks, which is important in modeling real-world processes!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Algebra
Formulas
Left-hand limit: lim(x → a-) = 3a + 4
Right-hand limit: lim(x → a+) = a + 3
Continuity condition: 3a + 4 = a + 3
Theorems
Continuity at a Point
Limit Theorems
Suitable Grade Level
Grades 10-12
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