Math Problem Statement
Find the values of constants a and b such that the function f(x) is continuous for all real numbers.
Solution
The given function is defined as follows:
\frac{x^2 + ax - 12}{x - 3} & \text{for } x \neq 3 \\ b & \text{for } x = 3 \end{cases}$$ We are asked to find the values of constants $$a$$ and $$b$$ that make $$f(x)$$ continuous for all real values of $$x$$. ### Step-by-Step Solution 1. **Continuity at $$x = 3$$:** For $$f(x)$$ to be continuous at $$x = 3$$, the following condition must be met: $$\lim_{x \to 3} f(x) = f(3) = b.$$ This means that the limit of $$f(x)$$ as $$x$$ approaches 3 (from the form $$\frac{x^2 + ax - 12}{x - 3}$$) must equal $$b$$. 2. **Simplifying the Expression for $$x \neq 3$$:** Let's simplify the expression $$\frac{x^2 + ax - 12}{x - 3}$$. We need to factor $$x^2 + ax - 12$$ so that it can be divided by $$x - 3$$: $$x^2 + ax - 12 = (x - 3)(x + c),$$ where we need to determine $$c$$ and $$a$$ such that this factorization holds. 3. **Finding $$a$$:** Expanding $$(x - 3)(x + c)$$ gives: $$x^2 + cx - 3x - 3c.$$Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Limits
Factorization of polynomials
Formulas
Limit definition for continuity: \( \lim_{x \to c} f(x) = f(c) \)
Factorization of quadratic expressions
Theorems
Continuity at a point
Suitable Grade Level
Grades 10-12