Use the graphs of the rational functions in choices A–D to answer each question.

There may be more than one correct choice. Which choices have domain (-∞, 3)U(3, ∞)?

Verified step by step guidance

1. Understand that the domain of a rational function consists of all real numbers except where the denominator is zero, as division by zero is undefined.
2. Identify the value(s) that make the denominator zero by setting the denominator equal to zero and solving for the variable. In this problem, the domain excludes the value 3, so the denominator must be zero at x = 3.
3. Examine each graph (choices A–D) to see if there is a vertical asymptote or a hole at x = 3, which indicates the function is undefined at that point.
4. For each graph, confirm that the function is defined for all x-values except x = 3, meaning the domain is \((-\infty, 3) \cup (3, \infty)\).
5. Select all choices where the graph shows a discontinuity at x = 3 and is continuous everywhere else, matching the given domain.

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Rational Functions and Their Graphs

Rational functions are ratios of polynomials. Their graphs often have vertical asymptotes at values where the denominator is zero, indicating points excluded from the domain. Understanding these asymptotes helps identify domain restrictions.

Interval Notation and Union of Intervals

Interval notation expresses sets of real numbers. The union symbol (U) combines intervals, indicating all values in either interval. For example, (-∞, 3) U (3, ∞) means all real numbers except 3, which is excluded from the domain.