In Exercises 1–30, find the domain of each function. f(x) = x² - 2x - 15

Verified step by step guidance

1. The function given is a polynomial function: \( f(x) = x^2 - 2x - 15 \).
2. Polynomial functions are defined for all real numbers.
3. Therefore, the domain of the function \( f(x) = x^2 - 2x - 15 \) is all real numbers.
4. In interval notation, the domain is expressed as \((-\infty, \infty)\).
5. This means that you can substitute any real number for \( x \) in the function without any restrictions.

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 polynomial functions like f(x) = x² - 2x - 15, the domain typically includes all real numbers, as there are no restrictions such as division by zero or square roots of negative numbers.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The function f(x) = x² - 2x - 15 is a quadratic polynomial, which is a specific type of polynomial where the highest exponent of the variable is 2. These functions are continuous and defined for all real numbers.

Finding the Domain

Finding the domain of a function involves identifying any restrictions on the input values. For f(x) = x² - 2x - 15, since it is a polynomial, we check for any values that would make the function undefined. In this case, there are no such values, so the domain is all real numbers, denoted as (-∞, ∞).