In Exercises 1–30, find the domain of each function. g(x) = 3/(x-4)

Verified step by step guidance

1. Identify the function: \( g(x) = \frac{3}{x-4} \).
2. Recognize that the function is a rational function, which means it is a fraction with a polynomial in the denominator.
3. Understand that the domain of a function is the set of all possible input values (x-values) for which the function is defined.
4. Determine that the function is undefined when the denominator is zero, so set the denominator equal to zero: \( x - 4 = 0 \).
5. Solve the equation \( x - 4 = 0 \) to find the value of \( x \) that makes the denominator zero, and exclude this value from the domain.

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. Understanding the domain is crucial because it determines the values that can be substituted into the function without resulting in undefined expressions, such as division by zero.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of g(x) = 3/(x-4), it is important to identify the values of x that make the denominator zero, as these values are excluded from the domain, leading to undefined behavior in the function.

Exclusions in Domain

When determining the domain of a function, any values that cause the function to be undefined must be excluded. For g(x) = 3/(x-4), the value x = 4 makes the denominator zero, thus it must be excluded from the domain, which is expressed as all real numbers except x = 4.