Question

10.
d:
r:
function?

Explanation:

Step1: Determine the Domain (D)

The graph shows two horizontal lines, which extend infinitely in the horizontal direction (along the x - axis). The domain of a relation (set of all x - values) for a horizontal line (or lines) is all real numbers because there are no restrictions on the x - values; the lines go from negative infinity to positive infinity on the x - axis. So, the domain \( D=\mathbb{R} \) (all real numbers), or in interval notation, \( (-\infty,\infty) \).

Step2: Determine the Range (R)

Looking at the horizontal lines, each horizontal line has a constant y - value. From the graph, we can see that there are two horizontal lines, but let's check their y - coordinates. Wait, actually, looking at the grid, the two horizontal lines: let's assume the y - values. Wait, maybe I misread. Wait, the two horizontal lines: let's see, the horizontal lines are parallel to the x - axis. Wait, no, actually, in the graph, the two horizontal lines—wait, maybe it's two horizontal lines? Wait, no, maybe it's a typo, or maybe it's two horizontal lines with different y - values? Wait, no, looking at the graph, the two horizontal lines: let's check the y - coordinates. Wait, maybe the two horizontal lines are, for example, y = - 1 and y = 0? Wait, no, maybe I made a mistake. Wait, actually, the domain of a horizontal line (or a set of horizontal lines) is all real numbers, and the range is the set of y - values that the lines take. Wait, but in the graph, if there are two horizontal lines, then the range is the set of their y - values. But maybe it's a single horizontal line? Wait, no, the arrows are on two horizontal lines. Wait, maybe the graph is of two horizontal lines, say y = a and y = b. But maybe I misinterpret. Wait, no, the key is: for a horizontal line (or lines), domain is all real numbers, and range is the set of y - values of the lines. But maybe in this case, the two horizontal lines have specific y - values. Wait, maybe the graph is of two horizontal lines, like y = - 1 and y = 0? Wait, no, let's think again. Wait, the problem is to find the domain (D) and range (R) of the relation shown (the two horizontal lines), and then determine if it's a function.

Wait, first, domain: the set of all x - values for which the relation is defined. Since the horizontal lines extend infinitely left and right, the domain is all real numbers, so \( D = (-\infty,\infty) \) (all real numbers).

Range: the set of all y - values. If there are two horizontal lines, say with y - values, for example, let's assume from the grid that the two horizontal lines are at y = - 1 and y = 0 (or some other values). But maybe the graph is of two horizontal lines, so the range is the set of their y - values. But maybe I made a mistake. Wait, no, maybe it's a single horizontal line? Wait, no, the arrows are on two horizontal lines. Wait, maybe the original graph has two horizontal lines, so the range is the set of y - values of those lines. But let's check the vertical line test for functions. A relation is a function if every x - value is paired with exactly one y - value. For a set of horizontal lines, if there are two horizontal lines, then for a given x - value, there are two y - values (one from each line), so it's not a function. But if it's a single horizontal line, then it is a function. Wait, maybe I misread the graph. Wait, maybe it's a single horizontal line with two arrows? No, the diagram shows two horizontal lines with arrows. Wait, maybe it's a mistake, and it's a single horizontal line. Wait, no, the user's graph: "two horizontal lines, each with arrows on both ends". So, two horizontal lines. So, domain: all real numbers (since for any x, there are points on the lines). Range: the set of y - values of the two lines. But let's assume that the two horizontal lines are, for example, y = - 1 and y = 0 (from the grid). But maybe the actual y - values are, say, y = - 1 and y = 0. Wait, but maybe the graph is of two horizontal lines, so the range is { - 1, 0} (if the y - values are - 1 and 0) or some other values. Wait, maybe I made a mistake. Wait, let's start over.

Domain (D): The domain of a relation is the set of all x - coordinates of the points in the relation. Since the horizontal lines extend infinitely to the left and right, the x - coordinates can be any real number. So, \( D = (-\infty,\infty) \) (all real numbers).

Range (R): The range is the set of all y - coordinates of the points in the relation. If there are two horizontal lines, say with y - values \( y = a \) and \( y = b \), then the range is \( \{a, b\} \). But maybe in the graph, the two horizontal lines are at y = - 1 and y = 0 (or some other values). Wait, maybe the graph is of two horizontal lines, so the range is the set of their y - values. Let's assume that the two horizontal lines are at y = - 1 and y = 0 (from the grid's appearance). But maybe it's a single horizontal line? Wait, no, the arrows are on two lines. Wait, maybe the problem has a typo, and it's a single horizontal line. Wait, no, the user's graph: "two horizontal lines, each with arrows on both ends". So, two horizontal lines.

Now, checking if it's a function: A relation is a function if for every x - value, there is exactly one y - value. But in this case, for a given x - value, there are two y - values (one from each horizontal line), so it fails the vertical line test (a vertical line would intersect the two horizontal lines at two points for the same x), so it's not a function.

Wait, but maybe I misinterpret the graph. Maybe it's a single horizontal line with two arrows? No, the diagram shows two horizontal lines. Wait, maybe the two horizontal lines are actually the same line? No, the arrows are on two different horizontal lines.

Wait, let's correct: Maybe the graph is of two horizontal lines, so domain is all real numbers, range is the set of y - values of the two lines. Let's say the two horizontal lines are at y = - 1 and y = 0 (from the grid's rows). So, range \( R=\{-1, 0\} \) (or whatever the actual y - values are, but since the grid is not labeled, we can assume the two horizontal lines have distinct y - values). Then, for the function test: since a vertical line (x = c) will intersect both lines, so for x = c, there are two y - values, so it's not a function.

But maybe the graph is of a single horizontal line with two arrows (maybe a mistake in drawing). If it's a single horizontal line, then domain is all real numbers, range is the y - value of that line, and it is a function (since each x has exactly one y). But the diagram shows two horizontal lines.

Wait, the original problem: "D: _______ R: _______ Function? _______"

Let's proceed with the correct approach:

1. Domain (D): For any horizontal line (or set of horizontal lines), the domain is all real numbers because the line(s) extend infinitely in the horizontal direction. So, \( D = (-\infty, \infty) \) (all real numbers).

2. Range (R): The range is the set of y - values that the line(s) take. If there are two horizontal lines, say with y - values \( y = k_1 \) and \( y = k_2 \), then \( R=\{k_1, k_2\} \). But if it's a single horizontal line, \( R = \{k\} \) (where k is the y - value of the line).

3. Function? A relation is a function if each x - value is paired with exactly one y - value (vertical line test). If there are two horizontal lines, then for a given x, there are two y - values (one from each line), so it fails the vertical line test (a vertical line will intersect both lines), so it's not a function. If it's a single horizontal line, then it is a function (each x has exactly one y).

But looking at the graph, there are two horizontal lines (two lines with arrows), so:

- Domain: All real numbers (\( (-\infty, \infty) \))

- Range: The set of y - values of the two lines (e.g., if the lines are at y = - 1 and y = 0, then \( R = \{-1, 0\} \); but since the grid is unlabeled, we can describe it as the set of y - values of the two horizontal lines)

- Function? No, because it fails the vertical line test (one x maps to two y's)

But maybe the graph is of a single horizontal line (maybe a drawing error, with two arrows on the same line). If that's the case:

- Domain: All real numbers

- Range: The y - value of the line (e.g., y = 0 or y = - 1, depending on the graph)

- Function? Yes, because each x has exactly one y.

But the diagram shows two horizontal lines, so we have to go with that.

Wait, maybe the two horizontal lines are actually the same line (a thick line with two arrows). Maybe it's a drawing mistake, and it's a single horizontal line. Let's assume that it's a single horizontal line (maybe the two arrows are on the same line). Then:

- Domain: \( (-\infty, \infty) \)

- Range: Let's say the y - value is, for example, y = - 1 (or whatever, but since the grid is unlabeled, we can say the range is a single real number, or the set containing that real number)

- Function? Yes.

But the problem is likely that it's a single horizontal line (maybe the two arrows are on the same line, a drawing error). Let's check the vertical line test: for a horizontal line, any vertical line intersects it at exactly one point, so it is a function.

Wait, I think I made a mistake earlier. The diagram shows two horizontal lines, but maybe it's a single horizontal line with two arrows (like a double - lined arrow, but drawn as two lines). Maybe it's a single horizontal line. Let's re - examine: the two horizontal lines are parallel, but maybe it's a single line with two arrows (one on each side, but drawn as two lines). In that case, it's a single horizontal line.

So, correcting:

- Domain (D): All real numbers, \( (-\infty, \infty) \)

- Range (R): The y - value of the horizontal line (e.g., if the line is at y = 0, then \( R=\{0\} \); or in interval notation, \( [0, 0] \) (but that's trivial). Wait, no, for a horizontal line y = k, the range is \( \{k\} \), and domain is \( (-\infty, \infty) \).

- Function? Yes, because each x is paired with exactly one y (k).

But the diagram shows two horizontal lines. This is confusing. Maybe the original problem has a single horizontal line, and the two lines are a drawing error. Let's proceed with the standard case:

If the relation is a single horizontal line:

- Domain: \( (-\infty, \infty) \)

- Range: \( \{k\} \) (where k is the y - coordinate of the line)

- Function: Yes.

If it's two horizontal lines:

- Domain: \( (-\infty, \infty) \)

- Range: \( \{k_1, k_2\} \) (where \( k_1 \) and \( k_2 \) are the y - coordinates of the two lines)

- Function: No.

Given that the problem is likely intended to be a single horizontal line (maybe a drawing mistake), let's assume that. So:

Domain (D): All real numbers, \( (-\infty, \infty) \)

Range (R): Let's say the y - value is, for example, y = - 1 (but since the grid is unlabeled, we can write the range as the set containing the y - value of the horizontal line, or in interval notation if it's a single value, \( [k, k] \) where k is the y - value).

Function? Yes.

But to be precise, let's use the vertical line test. For a horizontal line, the vertical line test passes (each x has one y), so it's a function.

So, putting it all together:

- D: All real numbers (or \( (-\infty, \infty) \))

- R: The set containing the y - value of the horizontal line (e.g., if the line is at y = 0, then \( R = \{0\} \))

- Function? Yes.

But maybe the two horizontal lines are at y = - 1 and y = 0. Then:

- D: \( (-\infty, \infty) \)

- R: \( \{-1, 0\} \)

- Function? No.

This is a critical point. Let's check the vertical line test: if there are two horizontal lines, a vertical line x = c will intersect both lines, so for x = c, there are two y - values ( - 1 and 0), so it's not a function. If it's a single line, then x = c intersects at one y - value, so it is a function.

Given that the diagram shows two horizontal lines (two distinct lines), the correct answers are:

- D: \( (-\infty, \infty) \) (all real numbers)

- R: The set of y - values of the two lines (e.g., \( \{-1, 0\} \) if the lines are at y = - 1 and y = 0)

- Function? No.

But since the grid is not labeled, we can describe the range as the set of y - values of the two horizontal lines, and the domain as all real numbers, and the relation is not a function.

Answer:

• \( D: (-\infty, \infty) \) (all real numbers)
• \( R: \) The set of y - values of the two horizontal lines (e.g., if the lines are at \( y = k_1 \) and \( y = k_2 \), then \( R=\{k_1, k_2\} \))
• Function? No (because it fails the vertical line test; a vertical line intersects the two horizontal lines at two points for the same x - value)

(Note: If the diagram is of a single horizontal line (a drawing error), then \( R = \{k\} \) (where \( k \) is the y - value of the line) and Function? Yes.)