In Exercises 76–81, find the domain of each function. g(x) = 4/(x - 7)

Verified step by step guidance

1. Identify the function: \( g(x) = \frac{4}{x - 7} \).
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) that will not cause the function to be undefined.
4. Determine when the denominator is zero, as this will make the function undefined. Set the denominator equal to zero: \( x - 7 = 0 \).
5. Solve the equation \( x - 7 = 0 \) to find the value of \( x \) that is not in the domain. The domain of \( g(x) \) is all real numbers except this value.

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) = 4/(x - 7), the numerator is a constant (4) and the denominator is a linear polynomial (x - 7). Understanding the structure of rational functions is essential for determining their domains.

Finding Restrictions

To find the domain of a function, one must identify any restrictions on the input values. For g(x) = 4/(x - 7), the restriction arises from the denominator, which cannot equal zero. Thus, solving the equation x - 7 = 0 reveals that x cannot be 7, leading to the conclusion that the domain excludes this value.