Math Problem Statement
determine if f is continous at the given values
Determine if the function f(x) is continuous at the given values x = -1 and x = 10.
Solution
To determine if the function is continuous at the given values, and , let's analyze the piecewise function provided in the image:
\begin{cases} \frac{x^2 + 5x + 4}{x^2 - 3x + 2} & \text{for } x \neq -1 \\ 3 & \text{for } x = -1 \end{cases}$$ ### 1. Continuity at $$x = -1$$: For the function to be continuous at $$x = -1$$, the following must hold: $$\lim_{x \to -1} f(x) = f(-1)$$ First, we calculate the limit as $$x$$ approaches $$-1$$ using the first case of the function: $$\lim_{x \to -1} \frac{x^2 + 5x + 4}{x^2 - 3x + 2}$$ We simplify the expression inside the limit: - The numerator $$x^2 + 5x + 4$$ can be factored as $$(x + 4)(x + 1)$$. - The denominator $$x^2 - 3x + 2$$ can be factored as $$(x - 1)(x - 2)$$. So the expression becomes: $$\frac{(x + 4)(x + 1)}{(x - 1)(x - 2)}$$ Evaluating the limit as $$x$$ approaches $$-1$$: $$\lim_{x \to -1} \frac{(x + 4)(x + 1)}{(x - 1)(x - 2)} = \frac{(-1 + 4)(-1 + 1)}{(-1 - 1)(-1 - 2)} = \frac{3 \times 0}{(-2) \times (-3)} = 0$$ Now, $$f(-1) = 3$$ (from the second case of the function). Since $$\lim_{x \to -1} f(x) = 0$$ and $$f(-1) = 3$$, the function is **not continuous** at $$x = -1$$. ### 2. Continuity at $$x = 10$$: To check for continuity at $$x = 10$$, we only need to ensure that the limit and the function value at $$x = 10$$ are equal because $$x = 10$$ falls under the first case of the piecewise function. $$f(10) = \frac{10^2 + 5(10) + 4}{10^2 - 3(10) + 2} = \frac{100 + 50 + 4}{100 - 30 + 2} = \frac{154}{72} = \frac{77}{36}$$ Since there is no discontinuity in the expression at $$x = 10$$, the function is **continuous** at $$x = 10$$. ### Conclusion: - **At $$x = -1$$:** The function is **not continuous**. - **At $$x = 10$$:** The function is **continuous**. Would you like more details or have any questions? Here are some related questions: 1. What does it mean for a function to be continuous at a point? 2. How do you factor quadratic expressions like $$x^2 + 5x + 4$$? 3. What is the significance of the limit in determining continuity? 4. How do you calculate limits involving rational functions? 5. What are common types of discontinuities in functions? **Tip:** Always check the limit from both sides when determining continuity at a specific point, especially for piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Rational Functions
Piecewise Functions
Formulas
f(x) = (x^2 + 5x + 4) / (x^2 - 3x + 2) for x ≠ -1
f(x) = 3 for x = -1
Limit formula: lim(x → c) f(x) = L
Theorems
Definition of Continuity
Limit Theorem for Rational Functions
Suitable Grade Level
Grades 11-12