In Exercises 1–30, find the domain of each function. f(x) = 1/(x+7) + 3/(x-9)

Verified step by step guidance

1. Identify the function: \( f(x) = \frac{1}{x+7} + \frac{3}{x-9} \).
2. Recognize that the domain of a function is the set of all possible input values (x-values) that will not cause the function to be undefined.
3. Determine where the function is undefined: This occurs when any denominator is equal to zero.
4. Set each denominator equal to zero and solve for \( x \): \( x+7 = 0 \) and \( x-9 = 0 \).
5. Solve these equations to find the values of \( x \) that are not in the domain: \( x = -7 \) and \( x = 9 \).

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 rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Understanding the domain is crucial for determining where the function can be evaluated.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 1/(x+7) + 3/(x-9), each term is a rational expression. The behavior of rational functions is significantly influenced by their denominators, which can introduce restrictions on the domain.

Finding Restrictions in the Domain

To find the domain of a function, one must identify values that cause the denominator to equal zero. For the function f(x) = 1/(x+7) + 3/(x-9), we set the denominators (x+7) and (x-9) to zero, leading to x = -7 and x = 9 as restrictions. Thus, the domain excludes these values, allowing us to express the domain in interval notation.