Math Problem Statement

Find all values of k that would make the function f(x) continuous at x = 4. f(x) = (x^2 - x - 12) / (x^2 - 3x - 4) for x ≠ 4 and k for x = 4.

Solution

To solve the given problem, we need to make the function f(x)f(x) continuous at x=4x = 4. The function f(x)f(x) is defined as:

\frac{x^2 - x - 12}{x^2 - 3x - 4} & \text{for } x \neq 4 \\ k & \text{for } x = 4 \end{cases}$$ ### Step 1: Simplify the function for $$x \neq 4$$ First, simplify the expression $$\frac{x^2 - x - 12}{x^2 - 3x - 4}$$. - Factor the numerator: $$x^2 - x - 12 = (x - 4)(x + 3)$$ - Factor the denominator: $$x^2 - 3x - 4 = (x - 4)(x + 1)$$ So, the function becomes: $$f(x) = \frac{(x - 4)(x + 3)}{(x - 4)(x + 1)} \quad \text{for} \quad x \neq 4$$ Cancel out $$(x - 4)$$ since it is present in both the numerator and denominator: $$f(x) = \frac{x + 3}{x + 1} \quad \text{for} \quad x \neq 4$$ ### Step 2: Ensure continuity at $$x = 4$$ To make the function continuous at $$x = 4$$, the following condition must hold: $$\lim_{x \to 4} f(x) = f(4)$$ Now, compute the limit of $$f(x)$$ as $$x$$ approaches 4: $$\lim_{x \to 4} f(x) = \lim_{x \to 4} \frac{x + 3}{x + 1} = \frac{4 + 3}{4 + 1} = \frac{7}{5}$$ ### Step 3: Set $$k = \lim_{x \to 4} f(x)$$ For continuity, $$f(4) = k$$, so we must have: $$k = \frac{7}{5}$$ Thus, the value of $$k$$ that makes the function continuous at $$x = 4$$ is: $$k = \frac{7}{5}$$ Would you like more details or have any questions? Here are some related questions to further explore the topic: 1. What does it mean for a function to be continuous at a point? 2. How do you handle cases where the numerator and denominator cancel out in rational functions? 3. What are the conditions for a piecewise function to be continuous at a specific point? 4. How can limits be used to determine the continuity of a function? 5. What are some real-life applications of continuous functions in calculus? **Tip**: Always check if you can factor and simplify rational functions, as this can help resolve potential discontinuities in the function.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity of Functions
Rational Functions
Factorization

Formulas

f(x) = (x^2 - x - 12) / (x^2 - 3x - 4) for x ≠ 4
k = lim (x -> 4) f(x)

Theorems

Limit Theorem
Continuity Theorem

Suitable Grade Level

Grade 11-12 or Calculus