In Exercises 1–30, find the domain of each function. g(x) = 2/(x+5)

Verified step by step guidance

1. Identify the function: \( g(x) = \frac{2}{x+5} \).
2. Recognize that the domain of a function includes all real numbers except where the function is undefined.
3. For rational functions like \( \frac{2}{x+5} \), the function is undefined where the denominator is zero.
4. Set the denominator equal to zero and solve for \( x \): \( x + 5 = 0 \).
5. Solve the equation: \( x = -5 \). Therefore, the domain of \( g(x) \) is all real numbers except \( x = -5 \).

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 typically restricted by values that would make the denominator zero, as division by zero is undefined.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of g(x) = 2/(x+5), the numerator is a constant (2) and the denominator is a linear polynomial (x+5). Understanding the structure of rational functions is essential for determining their domains.

Finding Restrictions on the Domain

To find the domain of a function, one must identify any restrictions that prevent certain x-values from being included. For g(x) = 2/(x+5), the restriction occurs when the denominator equals zero, leading to the equation x + 5 = 0, which must be solved to find the excluded value.