Find the domain of each function. g(x) = 3/(x²-2x-15)

Verified step by step guidance

1. Step 1: Recall that the domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like g(x) = 3/(x^2 - 2x - 15), the function is undefined when the denominator equals zero.
2. Step 2: Set the denominator equal to zero to find the x-values that make the function undefined. Solve the equation x^2 - 2x - 15 = 0.
3. Step 3: Factor the quadratic expression x^2 - 2x - 15. Look for two numbers that multiply to -15 and add to -2. These numbers are -5 and 3, so the factorization is (x - 5)(x + 3).
4. Step 4: Solve the factored equation (x - 5)(x + 3) = 0. Set each factor equal to zero: x - 5 = 0 and x + 3 = 0. Solve for x to find x = 5 and x = -3.
5. Step 5: Exclude the values x = 5 and x = -3 from the domain, as these make the denominator zero. The domain of g(x) is all real numbers except x = 5 and x = -3, which can be written in interval notation as (-∞, -3) ∪ (-3, 5) ∪ (5, ∞).

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. This is essential for identifying the values that make the denominator zero. In the case of g(x) = 3/(x^2-2x-15), factoring the quadratic will help determine the critical points that affect the domain.

Finding Asymptotes

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