Intermediate Algebra: Functions, Systems, and Polynomials Study Guide

Q3. Give the domain and range of the relation in interval notation. Decide whether the graph is a function and explain your answer.

Background

Topic: Functions and Graphs

This question tests your understanding of how to determine the domain and range from a graph, and how to decide if a graph represents a function using the vertical line test.

Key Terms and Concepts:

• Domain: The set of all possible input values (x-values) for the function.

• Range: The set of all possible output values (y-values) for the function.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Step-by-Step Guidance

1. Examine the graph to determine the leftmost and rightmost x-values where the graph exists. These values will help you write the domain in interval notation.

2. Look for any open or closed endpoints on the graph. A closed dot means the endpoint is included in the domain/range; an open dot means it is not included.

3. Identify the lowest and highest y-values the graph reaches to determine the range in interval notation.

4. Apply the vertical line test: Imagine drawing vertical lines across the graph. If any vertical line crosses the graph more than once, it is not a function.

5. Explain your reasoning for whether the graph is a function based on your observations.

Try solving on your own before revealing the answer!

Final Answer:

Domain: $[-3, 1]$ Range: $[-4, 1]$

Function? Yes, this graph is a function because it passes the vertical line test—no vertical line intersects the graph at more than one point.

The domain is from $x = -3$ to $x = 1$ (including both endpoints), and the range is from $y = -4$ to $y = 1$ (including both endpoints).