In Exercises 1–8, find the domain of each rational function. f(x)=(x+7)/(x^2+49)

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding rational functions is crucial for determining their properties, including their domain, which is influenced by the behavior of the denominator.

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 restricted by the values that make the denominator equal to zero, as division by zero is undefined. Identifying these restrictions is essential for accurately determining the domain.

Finding Restrictions

To find the domain of a rational function, one must identify the values of x that cause the denominator to equal zero. This involves solving the equation Q(x) = 0. The solutions to this equation represent the x-values that must be excluded from the domain, allowing for a complete understanding of where the function is valid.