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

Verified step by step guidance

1. Identify the rational function: $h(x)=\frac{x+7}{x^2-49}$.
2. Recognize that the domain of a rational function is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero to find the values that are not in the domain: $x^2-49=0$.
4. Factor the quadratic expression: $x^2-49=(x-7)(x+7)$.
5. Solve for $x$ in $(x-7)(x+7)=0$ to find the values to exclude from the domain.

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.