To determine if the graph is that of a function using the vertical-line test, we need to analyze the graph described.

Analysis of the graph:

• The graph contains two branches:
  1. One opens to the right starting from $(5, 0)$, passing through $(12.0, 6.6)$ and $(12.0, -6.6)$.
  2. The other opens to the left starting from $(-5, 0)$, passing through $(-12.0, -6.6)$ and $(-12.0, 6.6)$.

From this, we can visualize two curves: one starting at $(5,0)$ and moving right, and another starting at $(-5,0)$ and moving left. Each curve passes through points on both the positive and negative sides of the $y$-axis at the same $x$-coordinate (e.g., $x = 12$ and $x = -12$).

Applying the vertical-line test:

• A vertical line at $x = 12$ intersects the graph at two points (i.e., one at $(12.0, 6.6)$ and one at $(12.0, -6.6)$).
• Similarly, a vertical line at $x = -12$ intersects the graph at two points.

Since a vertical line intersects the graph at more than one point at certain values of $x$, the graph is not a function.

Correct answer:

A. No, the graph is not a function because a vertical line $x = 7$ intersects the graph at two points.

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(a) Domain and Range:

• Domain: The graph includes $x$-values from $-12$ to $12$, but not all values between these two points. Specifically, there are no values for $x$ between $-5$ and $5$ (in the interior of the graph). Therefore, the domain is $[-12, -5] \cup [5, 12]$.
• Range: The $y$-values vary from approximately $-6.6$ to $6.6$ for both branches. So, the range is $[-6.6, 6.6]$.

(b) The intercepts:

• x-intercepts: The graph intersects the $x$-axis at $x = -5$ and $x = 5$. These are the points $(-5, 0)$ and $(5, 0)$.
• y-intercepts: There are no y-intercepts because the graph never crosses the $y$-axis.

(c) Symmetry:

• Symmetry with respect to the y-axis: The graph is not symmetric with respect to the $y$-axis because the left and right branches do not mirror each other along the axis.
• Symmetry with respect to the x-axis: The graph is symmetric with respect to the $x$-axis because the left branch's points $(-12.0, -6.6)$ and $(-12.0, 6.6)$ mirror the points $(12.0, -6.6)$ and $(12.0, 6.6)$.
• Symmetry with respect to the origin: The graph is not symmetric with respect to the origin because the left and right branches do not form a perfect rotational symmetry.

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Would you like any further clarification on these concepts, or are there any other questions you'd like to explore? Here are some related topics you might find interesting:

1. What is the importance of the vertical-line test in determining functions?
2. How does symmetry help in graphing functions and understanding their behavior?
3. What are the differences between a function and a relation in mathematics?
4. How do we determine the domain and range of more complex functions?
5. Can a function still be continuous if it fails the vertical-line test?

Tip: The vertical-line test is an essential tool for distinguishing functions from relations. A function must pass this test, meaning that for any $x$-value, there is only one corresponding $y$-value.