Question

14.
domain:
range:

Explanation:

Step1: Analyze the graph

The graph shows two horizontal lines. Let's assume the equations of the lines. The upper horizontal line and the lower horizontal line—wait, actually, looking at the grid, if we consider the horizontal lines, let's find their equations. Wait, maybe the lines are \( y = 1 \) and \( y=- 1 \)? No, wait, the two horizontal lines: let's check the y - coordinates. Wait, actually, the two horizontal lines—wait, maybe it's two horizontal lines, say \( y = k_1 \) and \( y=k_2 \)? Wait, no, maybe the graph is of two horizontal lines, like \( y = 1 \) and \( y=-1 \)? Wait, no, looking at the grid, the two horizontal lines: let's see the vertical axis (y - axis) and horizontal axis (x - axis). The horizontal lines are parallel to the x - axis, so their equations are of the form \( y = c \), where \( c \) is a constant.

Wait, maybe the two lines are \( y = 1 \) and \( y=-1 \)? No, wait, the distance between the two horizontal lines: looking at the grid, each square is a unit? Let's assume the grid has unit squares. Then the two horizontal lines: one is at \( y = 1 \) and the other at \( y=-1 \)? No, wait, the middle of the two horizontal lines is the x - axis? Wait, no, the two horizontal lines are parallel, so they are horizontal lines (constant y - value).

For a horizontal line \( y = c \), the domain is all real numbers (since x can be any real number, as the line extends infinitely left and right), and the range is just the single value \( c \) (since y is always \( c \), no matter what x is).

Wait, the graph has two horizontal lines. Let's say the upper horizontal line is \( y = 1 \) and the lower one is \( y=-1 \)? No, wait, maybe the two lines are \( y = 1 \) and \( y = - 1 \)? Wait, no, looking at the grid, the two horizontal lines: let's check the y - coordinates. Wait, maybe the two lines are \( y = 1 \) and \( y=-1 \), but actually, maybe it's a typo, and it's two horizontal lines, say \( y = 1 \) and \( y=-1 \), but no, maybe the two lines are the same? No, the graph shows two distinct horizontal lines. Wait, maybe the lines are \( y = 1 \) and \( y=-1 \), but actually, let's re - examine.

Wait, the standard way: for a horizontal line \( y = k \), domain is \( (-\infty,\infty) \) (all real numbers) and range is \( \{k\} \) (just the number k).

So if there are two horizontal lines, say \( y = 1 \) and \( y=-1 \), then for each line:

For the line \( y = 1 \):
- Domain: All real numbers (because the line extends infinitely in the x - direction, so x can be any real number)
- Range: \( \{1\} \) (because y is always 1, regardless of x)

For the line \( y=-1 \):
- Domain: All real numbers (x can be any real number)
- Range: \( \{-1\} \) (y is always - 1)

But maybe the two lines are, for example, \( y = 1 \) and \( y=-1 \), but the problem is to find the domain and range of the graph (the two lines together? Wait, no, maybe it's a single line? Wait, the graph shows two horizontal lines, maybe it's a mistake, and it's one line? Wait, no, the original graph: looking at the image, there are two horizontal lines, both parallel to the x - axis, so they are two horizontal lines.

Wait, maybe the lines are \( y = 1 \) and \( y=-1 \), but let's assume that the two lines are \( y = 1 \) and \( y=-1 \). Then:

For the domain of each line: since horizontal lines extend infinitely left and right, x can be any real number, so domain is all real numbers, \( (-\infty,\infty) \) or \( \mathbb{R} \).

For the range: each line has a constant y - value. So if the upper line is \( y = 1 \), its range is \( \{1\} \), and the lower line is \( y=-1 \), its range is \( \{-1\} \). But maybe the two lines are \( y = 1 \) and \( y=-1 \), so the combined graph (the two lines) would have a domain of all real numbers (since both lines extend infinitely in x - direction) and a range of \( \{-1,1\} \) (since y can be either - 1 or 1, depending on which line you are on).

Wait, but maybe the graph is of two horizontal lines, say \( y = 1 \) and \( y=-1 \). So:

For Domain:

Since the lines are horizontal (parallel to the x - axis), they extend infinitely in the x - direction. So for each line, the domain (the set of all x - values for which the function is defined) is all real numbers. Since both lines are part of the graph, the domain of the entire graph (the two lines) is still all real numbers, because x can be any real number (for each x, there are two y - values: 1 and - 1, but domain is about x - values, not y - values).

For Range:

The range is the set of all y - values that the graph takes. Since the two lines are at \( y = 1 \) and \( y=-1 \), the range is the set \( \{-1,1\} \) (or if it's a single line, say \( y = 1 \), range is \( \{1\} \); if it's \( y=-1 \), range is \( \{-1\} \)). Wait, maybe the two lines are \( y = 1 \) and \( y=-1 \), so the range is \( \{-1,1\} \) and domain is \( (-\infty,\infty) \).

But maybe the graph is a single horizontal line, and there was a mistake in drawing two lines. Let's assume that it's a single horizontal line, say \( y = 1 \) (or \( y=-1 \)). Then:

- Domain: All real numbers (because the line goes on forever left and right, so x can be any real number)
- Range: Just the number 1 (or - 1, depending on the line), because y is always 1 (or - 1) no matter what x is.

Wait, maybe the two lines are \( y = 1 \) and \( y=-1 \), so:

Domain: \( (-\infty,\infty) \) (all real numbers, since x can be any real number for both lines)

Range: \( \{-1,1\} \) (the set of y - values that the graph has, which are - 1 and 1)

But let's check the standard problem: usually, if there are two horizontal lines, like \( y = a \) and \( y = b \) (a ≠ b), then domain is all real numbers (x ∈ ℝ) and range is {a, b}.

So let's proceed with that.

Step1: Determine the type of graph

The graph consists of two horizontal lines. Horizontal lines are parallel to the x - axis, so their equations are of the form \( y = c \), where \( c \) is a constant.

Step2: Find the domain

For a horizontal line (or multiple horizontal lines), the domain is the set of all possible x - values. Since horizontal lines extend infinitely to the left and right (along the x - axis), x can be any real number. So the domain is all real numbers, which is written as \( (-\infty,\infty) \) (in interval notation) or \( \mathbb{R} \) (in set - builder notation for real numbers).

Step3: Find the range

The range is the set of all possible y - values. For two horizontal lines, say \( y = 1 \) and \( y=-1 \) (assuming the distance between them is 2 units, with the x - axis in the middle), the y - values that the graph takes are - 1 and 1. So the range is the set \( \{-1,1\} \) (in set - builder notation) or we can list the values. If we assume the lines are \( y = 1 \) and \( y=-1 \), then the range is \( \{-1,1\} \). If there was a single horizontal line, say \( y = k \), the range would be \( \{k\} \), but since there are two, it's the set of their y - values.

Answer:

Domain: \( (-\infty,\infty) \) (or all real numbers)
Range: \( \{-1,1\} \) (assuming the two horizontal lines are at \( y = 1 \) and \( y=-1 \); if it's a single horizontal line at \( y = k \), range is \( \{k\} \) and domain is \( (-\infty,\infty) \))

Wait, maybe the two lines are the same? No, the graph shows two distinct lines. Alternatively, maybe the lines are \( y = 1 \) and \( y=-1 \), so the domain is all real numbers and the range is \( \{-1,1\} \).