Q23. Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find:

• (a) Is the graph a function?

• (b) The domain and range

• (c) The intercepts, if any

• (d) Any symmetry with respect to the x-axis, the y-axis, or the origin

Background

Topic: Functions and Graphs

This question tests your understanding of how to determine if a graph represents a function (using the vertical-line test), and how to extract key properties such as domain, range, intercepts, and symmetry from a graph.

Key Terms and Concepts:

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

• 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.

• Intercepts: Points where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercepts).

• Symmetry: A graph can be symmetric with respect to the x-axis, y-axis, or origin.

Step-by-Step Guidance

1. Examine the graph and apply the vertical-line test: Imagine drawing vertical lines at various x-values. Does any vertical line cross the graph more than once?

2. Identify the domain: Look at the leftmost and rightmost x-values where the graph exists. List all x-values for which the graph is defined.

3. Identify the range: Observe the lowest and highest y-values the graph attains. List all y-values that the graph reaches.

4. Find the intercepts: Look for points where the graph crosses the x-axis (set y = 0) and the y-axis (set x = 0).

5. Check for symmetry: Fold the graph along the x-axis, y-axis, and origin. Does the graph match itself in any of these cases?

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