In Exercises 1–30, find the domain of each function. g(x) = 3/(x^2-2x-15)

Verified step by step guidance

1. Identify the function: \( g(x) = \frac{3}{x^2 - 2x - 15} \).
2. Recognize that the domain of a function is the set of all possible input values (x-values) for which the function is defined.
3. For rational functions, the function is undefined where the denominator is zero. Therefore, set the denominator equal to zero: \( x^2 - 2x - 15 = 0 \).
4. Solve the quadratic equation \( x^2 - 2x - 15 = 0 \) by factoring. Look for two numbers that multiply to -15 and add to -2.
5. Once factored, set each factor equal to zero and solve for \( x \). These values are the points where the function is undefined, and thus should be excluded from the domain.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that would make the denominator zero, as division by zero is undefined.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting them as a product of their linear factors. For the function g(x) = 3/(x^2 - 2x - 15), factoring the quadratic in the denominator helps identify the values of x that make the denominator zero, which are critical for determining the domain.

Identifying Asymptotes

Asymptotes are lines that a graph approaches but never touches. In the context of rational functions, vertical asymptotes occur at the x-values that make the denominator zero. Understanding these points is essential for determining the domain, as they indicate where the function is undefined.