In Exercises 1–8, find the domain of each rational function. h(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. In the case of h(x) = (x + 7) / (x^2 - 49), the numerator is a linear polynomial and the denominator is a quadratic polynomial. Understanding the structure of rational functions is essential for analyzing their properties, including their domain.

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 zero, as division by zero is undefined. Identifying these restrictions is crucial for determining the domain of h(x).

Finding Restrictions

To find the domain of a rational function, one must identify the values of x that cause the denominator to equal zero. For h(x), this involves solving the equation x^2 - 49 = 0, which factors to (x - 7)(x + 7) = 0. The solutions, x = 7 and x = -7, indicate the points where the function is undefined, thus defining the domain.