Find the domain of each function. g(x) = 4/(x - 7)

Verified step by step guidance

1. Identify the function given: \(g(x) = \frac{4}{x - 7}\).
2. Recall that the domain of a function includes all real numbers except those that make the denominator zero, because division by zero is undefined.
3. Set the denominator equal to zero to find values to exclude: \(x - 7 = 0\).
4. Solve the equation for \(x\): \(x = 7\).
5. Conclude that the domain of \(g(x)\) is all real numbers except \(x = 7\), which can be written in interval notation as \((-\infty, 7) \cup (7, \infty)\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. It excludes any values that cause undefined expressions, such as division by zero or taking the square root of a negative number in the real number system.

Restrictions from Denominators

When a function includes a fraction, the denominator cannot be zero because division by zero is undefined. To find the domain, identify values of x that make the denominator zero and exclude them from the domain.

Solving Equations to Find Domain Restrictions

To determine domain restrictions, set the denominator equal to zero and solve for x. The solutions are the values that must be excluded from the domain. For g(x) = 4/(x - 7), solve x - 7 = 0 to find the restricted value.