Math Problem Statement
Solution
To solve the problem of finding the value of that makes the function continuous, we need to ensure that the left-hand limit and right-hand limit of the function are equal at the point where the piecewise definition changes, which is .
The function is defined as:
\begin{cases} k^2 - 2x, & x < 9 \\ kx, & x \geq 9 \end{cases}$$ For the function to be continuous at $$x = 9$$: 1. The left-hand limit as $$x$$ approaches 9 must equal the right-hand limit. 2. The value of the function at $$x = 9$$ must equal these limits. ### Step-by-Step Solution 1. **Left-hand limit as $$x \to 9^-$$:** $$\lim_{{x \to 9^-}} f(x) = k^2 - 2(9) = k^2 - 18$$ 2. **Right-hand limit as $$x \to 9^+$$:** $$\lim_{{x \to 9^+}} f(x) = k(9) = 9k$$ 3. **Setting the limits equal for continuity:** $$k^2 - 18 = 9k$$ 4. **Solving the quadratic equation:** Rewrite the equation: $$k^2 - 9k - 18 = 0$$ This is a standard quadratic equation of the form $$ax^2 + bx + c = 0$$, where $$a = 1$$, $$b = -9$$, and $$c = -18$$. Use the quadratic formula: $$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substituting the values of $$a$$, $$b$$, and $$c$$: $$k = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-18)}}{2(1)}$$ $$k = \frac{9 \pm \sqrt{81 + 72}}{2}$$ $$k = \frac{9 \pm \sqrt{153}}{2}$$ 5. **Calculating the possible values of $$k$$:** The two possible values of $$k$$ are: $$k_1 = \frac{9 + \sqrt{153}}{2}, \quad k_2 = \frac{9 - \sqrt{153}}{2}$$ These are the values of $$k$$ that make the function continuous. ### Conclusion The function $$f(x)$$ is continuous for the values of $$k$$: $$k = \frac{9 + \sqrt{153}}{2} \quad \text{or} \quad k = \frac{9 - \sqrt{153}}{2}$$ Would you like more details or have any questions? ### Related Questions: 1. How do you determine the continuity of a function at a specific point? 2. What is the significance of continuity in a real-world scenario? 3. How does a discontinuity affect the graph of a function? 4. Can you provide an example of a function that is continuous everywhere except at one point? 5. What are the differences between continuous and differentiable functions? ### Tip: Always remember to check both left-hand and right-hand limits when determining continuity at a point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Quadratic Equations
Formulas
Quadratic equation formula: ax^2 + bx + c = 0
Continuity condition: left-hand limit = right-hand limit at x = 9
Theorems
Continuity Theorem
Quadratic Formula
Suitable Grade Level
Grades 10-12
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