Question

find the domain and the range of the relation. also determine whether the relation is a function.
the domain is
(type each answer only once. use a comma to separate answers as needed)

Explanation:

Step1: Analyze the Graph

The graph appears to be a vertical line (since it's a straight line parallel to the y - axis). For a vertical line, the x - coordinate is constant for all points on the line. From the graph (assuming the vertical line is at \(x = a\), but looking at the grid, if we assume the vertical line is at \(x=0\) (the y - axis), but wait, no, a vertical line has the equation \(x = k\) where \(k\) is a constant. Wait, maybe the graph is a vertical line, say \(x = 0\) (the y - axis) or some other constant x - value. Wait, the problem is about finding the domain, range, and whether it's a function.

Step2: Find the Domain

The domain of a relation is the set of all x - values (input values) for which the relation is defined. For a vertical line \(x = c\) (where \(c\) is a constant), the x - value is always \(c\). So the domain is just the set containing that single x - value. If the vertical line is at \(x = 0\) (the y - axis), the domain is \(\{0\}\).

Step3: Find the Range

The range of a relation is the set of all y - values (output values) that the relation takes. For a vertical line, as \(y\) can take any real number (since the line extends infinitely up and down), the range is all real numbers, \((-\infty,\infty)\) or \(\mathbb{R}\).

Step4: Determine if it's a Function

A relation is a function if for every x - value in the domain, there is exactly one y - value in the range. For a vertical line, a single x - value (from the domain) is paired with infinitely many y - values. So by the vertical line test (a graph represents a function if no vertical line intersects the graph more than once), a vertical line fails the vertical line test. So the relation is not a function.

But wait, the problem's graph is not fully visible. Wait, maybe the graph is a vertical line. Let's assume the vertical line is at \(x = 0\) (the y - axis). Then:

Domain: The set of x - values. For a vertical line \(x = 0\), the domain is \(\{0\}\).

Range: The set of y - values. Since the line goes from \(-\infty\) to \(\infty\) in the y - direction, the range is all real numbers, \((-\infty,\infty)\).

Function? No, because one x (0) is paired with multiple y's.

But maybe the graph is different. Wait, the user's image shows a vertical line (probably \(x = 0\), the y - axis) and a grid. Let's proceed with the standard vertical line case.

Answer:

Domain: \(\{0\}\), Range: \((-\infty,\infty)\), Not a function.

(Note: If the vertical line is at a different x - value, say \(x = 2\), the domain would be \(\{2\}\), range still \((-\infty,\infty)\), and not a function. The key is that vertical lines are not functions.)