Math Problem Statement
find the value of the constant a that makes the function continuous
(x)={(a^(2)+4x,x<-1),(ax+16,x>=-1):}
Solution
To determine the value of the constant that makes the function continuous, we require that the left-hand limit and the right-hand limit of the function at are equal to the function's value at .
The given function is:
\begin{cases} a^2 + 4x & \text{if } x < -1, \\ ax + 16 & \text{if } x \geq -1. \end{cases}$$ The condition for continuity at $$x = -1$$ is: $$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) = f(-1).$$ ### Step 1: Compute $$\lim_{x \to -1^-} f(x)$$ For $$x < -1$$, $$f(x) = a^2 + 4x$$. So: $$\lim_{x \to -1^-} f(x) = a^2 + 4(-1) = a^2 - 4.$$ ### Step 2: Compute $$\lim_{x \to -1^+} f(x)$$ For $$x \geq -1$$, $$f(x) = ax + 16$$. So: $$\lim_{x \to -1^+} f(x) = a(-1) + 16 = -a + 16.$$ ### Step 3: Equate the limits For the function to be continuous, the two limits must be equal: $$a^2 - 4 = -a + 16.$$ ### Step 4: Solve for $$a$$ Simplify the equation: $$a^2 + a - 20 = 0.$$ Factorize: $$(a + 5)(a - 4) = 0.$$ Thus: $$a = -5 \quad \text{or} \quad a = 4.$$ ### Step 5: Verify both solutions Both values of $$a$$ satisfy the condition for continuity. Therefore, the function is continuous for $$a = -5$$ or $$a = 4$$. --- Would you like a detailed explanation of why both values of $$a$$ work, or do you have further questions? ### Related Questions: 1. What is the definition of continuity at a point in a piecewise function? 2. How do you determine the limits for a piecewise function at a boundary point? 3. Can a piecewise function have more than one solution for continuity? 4. What happens if $$a$$ is not -5 or 4 in this case? 5. Can the second derivative also be continuous at $$x = -1$$ for some values of $$a$$? ### Tip: Always verify both the left-hand and right-hand limits to ensure a function is continuous at a boundary point in piecewise functions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Quadratic Equations
Formulas
Continuity condition: lim(x → c^-) f(x) = lim(x → c^+) f(x) = f(c)
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Limit definition of continuity
Properties of quadratic equations
Suitable Grade Level
Grades 10-12