Math Problem Statement
For what two integer values of a is the following function continuous at every x?
g(x) = {a * x^2 + 2a * x if x <= 1, 3 if x > 1}
Solution
The problem provided asks for the two integer values of that make the given function continuous at every . The piecewise function is:
\begin{cases} a x^2 + 2a x & \text{if } x \leq 1 \\ 3 & \text{if } x > 1 \end{cases}$$ To ensure the function is continuous at $$x = 1$$, the two pieces of the function must be equal at $$x = 1$$. ### Steps: 1. **Continuity Condition**: For the function to be continuous at $$x = 1$$, the limit of $$g(x)$$ as $$x$$ approaches 1 from the left must equal the limit as $$x$$ approaches 1 from the right. Thus, we set the two pieces equal at $$x = 1$$: $$a(1)^2 + 2a(1) = 3$$ 2. **Simplifying**: $$a + 2a = 3$$ $$3a = 3$$ $$a = 1$$ Thus, the value $$a = 1$$ ensures the function is continuous. ### Verification: To confirm, let's plug $$a = 1$$ back into the function. For $$x \leq 1$$, the function becomes: $$g(x) = 1x^2 + 2(1)x = x^2 + 2x$$ At $$x = 1$$: $$g(1) = 1^2 + 2(1) = 1 + 2 = 3$$ This matches the value of the function for $$x > 1$$, which is also 3. Therefore, the function is continuous for $$a = 1$$. ### Conclusion: The function is continuous for $$a = 1$$. Would you like further details or any additional questions answered on this topic? Here are five related questions for practice: 1. What does it mean for a function to be continuous at a point? 2. How do you verify if a function is continuous on an interval? 3. Can a function be continuous if it has different expressions on different intervals? 4. What role do limits play in determining continuity of a function? 5. How would the function change if it were discontinuous at $$x = 1$$? **Tip**: When solving for continuity, always check that the left-hand and right-hand limits match at the point of concern.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Algebra
Formulas
a * x^2 + 2a * x = 3 at x = 1
Theorems
Continuity Condition
Suitable Grade Level
Grades 10-12
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