this is the graph of a linear function.
what are the domain and the range of the function?
domain:
-2 ≤ x ≤ 2
-2

Question

this is the graph of a linear function.
what are the domain and the range of the function?
domain:
-2 ≤ x ≤ 2
-2 < x < 2
-6 ≤ x ≤ 6
-6 < x < 6
all real numbers
range:

Accepted Answer

# Explanation:
## Step1: Analyze the domain
The graph of the linear function has open circles at $ x = -2 $ (y - value 6) and $ x = 2 $ (y - value - 6), and the line is between these two x - values. The domain represents the set of all x - values for which the function is defined. Since the endpoints at $ x=-2 $ and $ x = 2 $ are not included (open circles), the domain is $ - 2<x<2 $? Wait, no, wait. Wait, looking at the graph, the left end is at $ x=-2 $ (open circle) and the right end is at $ x = 2 $ (open circle)? Wait, no, the x - coordinates of the endpoints: the left point is at $ x=-2 $ (open circle) and the right point is at $ x = 2 $ (open circle). Wait, but the options for domain: let's re - examine. Wait, the graph is a line segment (since it's a linear function with endpoints). Wait, the open circles mean that the points at $ x=-2 $ and $ x = 2 $ are not included. So the x - values start just above - 2 and end just below 2? No, wait, the x - coordinates of the two endpoints: the left endpoint is at $ x=-2 $ (open circle) and the right endpoint is at $ x = 2 $ (open circle). So the domain is the set of x - values between - 2 and 2, not including - 2 and 2? Wait, no, wait the options: one of the options is $ -2<x<2 $, another is $ -2\leq x\leq2 $. Wait, open circles mean that the endpoints are not included. So for the domain, the x - values are from - 2 (not included) to 2 (not included), so $ -2 < x<2 $? Wait, no, wait the graph: the left end is at $ x=-2 $ (open circle) and the right end is at $ x = 2 $ (open circle). So the domain is $ -2<x<2 $? Wait, but let's check the range.

## Step2: Analyze the range
The range is the set of all y - values. The minimum y - value on the graph is - 6 (at $ x = 2 $, open circle) and the maximum y - value is 6 (at $ x=-2 $, open circle). Since the endpoints are not included, the range should be $ -6 < y<6 $? Wait, no, wait the line is a linear function, let's find the equation. The line passes through the origin (0,0), and the two endpoints: (- 2,6) open circle and (2, - 6) open circle. The slope $ m=\frac{-6 - 0}{2-0}=\frac{-6}{2}=-3 $, and also $ m=\frac{6 - 0}{-2-0}=\frac{6}{-2}=-3 $. The equation of the line is $ y=-3x $. When $ x=-2 $, $ y = 6 $ (open circle), when $ x = 2 $, $ y=-6 $ (open circle). So the y - values go from just above - 6 (since $ x = 2 $ is not included, $ y=-3\times2=-6 $ is not included) to just below 6 (since $ x=-2 $ is not included, $ y=-3\times(-2) = 6 $ is not included). Wait, but the options for range? Wait, the options for domain: let's go back. The domain is the set of x - values. The graph is a line segment between $ x=-2 $ (open) and $ x = 2 $ (open), so the domain is $ -2<x<2 $? Wait, no, the first option for domain is $ -2\leq x\leq2 $, second is $ -2<x<2 $, third is $ -6\leq x\leq6 $, fourth is $ -6<x<6 $, fifth is all real numbers.

Wait, I made a mistake. Let's look at the x - axis. The leftmost x - value of the graph is $ x=-2 $ (open circle) and the rightmost x - value is $ x = 2 $ (open circle). So the domain is the set of x - values such that $ -2<x<2 $. Now for the range: the y - values. The lowest y - value is - 6 (at $ x = 2 $, open circle) and the highest y - value is 6 (at $ x=-2 $, open circle). So the range is the set of y - values such that $ -6<y<6 $? Wait, but let's check the options. Wait, the problem's domain options: let's re - check the graph. Wait, the x - coordinates of the two endpoints: the left point is at $ x=-2 $ (open circle) and the right point is at $ x = 2 $ (open circle). So the domain is $ -2<x<2 $. Wait, but the first option for domain is $ -2\leq x\leq2 $ (closed interval), second is $ -2<x<2 $ (open interval), third is $ -6\leq x\leq6 $, fourth is $ -6<x<6 $, fifth is all real numbers.

Wait, maybe I misread the graph. Wait, the x - axis: the grid lines are at integer values. The left end of the graph is at $ x=-2 $ (open circle) and the right end is at $ x = 2 $ (open circle). So the domain is $ -2<x<2 $. Now for the range: the y - values. The line goes from $ y = 6 $ (at $ x=-2 $, open circle) down to $ y=-6 $ (at $ x = 2 $, open circle). So the range is $ -6<y<6 $? But let's check the options. Wait, the problem's range options are not shown, but the domain: let's confirm. The graph is a linear function (a line segment) with endpoints at $ (-2,6) $ (open) and $ (2,-6) $ (open). So the domain (x - values) is from - 2 to 2, not including - 2 and 2, so $ -2<x<2 $. And the range (y - values) is from - 6 to 6, not including - 6 and 6, so $ -6<y<6 $. But let's check the domain options again. The options for domain are: $ -2\leq x\leq2 $, $ -2<x<2 $, $ -6\leq x\leq6 $, $ -6<x<6 $, all real numbers. So the correct domain is $ -2<x<2 $ (since the endpoints are open circles, so not included) and the range is $ -6<y<6 $ (since the y - values at the endpoints are not included). But the question is about domain and range. Wait, the user's question is to find the domain and range. Let's focus on the domain first. The graph has open circles at $ x=-2 $ and $ x = 2 $, so the domain is $ -2<x<2 $.

Wait, no, wait I think I messed up. Wait, the x - coordinates of the two points: the left point is at $ x=-2 $ (open circle) and the right point is at $ x = 2 $ (open circle). So the domain is the set of x - values between - 2 and 2, not including - 2 and 2, so $ -2<x<2 $.

## Step3: Analyze the range
The range is the set of y - values. The minimum y - value is - 6 (at $ x = 2 $, open circle) and the maximum y - value is 6 (at $ x=-2 $, open circle). So the range is $ -6<y<6 $. But let's check the options. Wait, the problem's range dropdown is not fully shown, but the domain options: the correct domain is $ -2<x<2 $? Wait, no, wait the first option for domain is $ -2\leq x\leq2 $ (closed), second is $ -2<x<2 $ (open). Since the circles are open, the endpoints are not included, so domain is $ -2<x<2 $.

# Answer:
Domain: $ -2 < x<2 $ (corresponding to the option $ -2 < x<2 $)
Range: $ -6 < y<6 $ (assuming the range options follow the same logic, with open circles at $ y = 6 $ and $ y=-6 $, so the range is $ -6 < y<6 $)

Wait, but let's re - check the domain. The graph is a line segment between $ x=-2 $ (open) and $ x = 2 $ (open), so the domain is $ -2<x<2 $. So the domain is $ -2 < x<2 $ (the option $ -2 < x<2 $) and the range is $ -6 < y<6 $.

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Explanation:

Step1: Analyze the domain

Step2: Analyze the range

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