In Exercises 1–8, find the domain of each rational function. g(x)=3x^2/(x−5)(x+4)

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 and Q are polynomials. Understanding rational functions is crucial for analyzing their behavior, particularly in terms of their domain and any restrictions that may arise from 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 typically all real numbers except where the denominator equals zero, as division by zero is undefined. Identifying these restrictions is essential for determining the valid inputs for the function.

Finding Restrictions

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