The given graph appears to be a curve. Let's analyze whether this relation is a function and determine its domain and range.

Step 1: Determine if it is a function

To check if a relation is a function, we apply the vertical line test. This test states that if any vertical line intersects the graph more than once, the relation is not a function. If every vertical line intersects the graph at most once, the relation is a function.

From the graph:

• Any vertical line drawn across the curve will intersect the graph at only one point.

Thus, the graph passes the vertical line test, meaning the relation is a function.

Step 2: Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• From the graph, the function exists for all values of $x$ starting from approximately $x = -2$ and continues indefinitely to the right.
• Thus, the domain is $x \in [-2, \infty)$.

Step 3: Range

The range of a function is the set of all possible output values (y-values).

• The graph starts from $y = -6$ and extends upwards without bound.
• Therefore, the range is $y \in [-6, \infty)$.

Summary:

• Relation: This is a function.
• Domain: $x \in [-2, \infty)$
• Range: $y \in [-6, \infty)$

Would you like further clarification, or do you have any questions? Here are five related questions to expand on this concept:

1. What is the significance of the vertical line test in determining if a relation is a function?
2. How would the domain change if the curve were shifted horizontally?
3. Can a function have a range that is bounded above but not below?
4. What happens to the function if the curve is reflected across the x-axis?
5. How does the concept of one-to-one functions relate to this graph?

Tip: Always check for horizontal asymptotes or breaks in the graph when determining the range, especially when the curve approaches infinity.