The problem asks us to determine if the given function, represented by discrete points on a graph, is linear and to describe its domain and range. Additionally, we need to describe how the y-values change with respect to x-values.
2025/4/18
1. Problem Description
The problem asks us to determine if the given function, represented by discrete points on a graph, is linear and to describe its domain and range. Additionally, we need to describe how the y-values change with respect to x-values.
2. Solution Steps
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6. A linear function has a constant rate of change (slope). Examining the graph, we can observe the coordinates of the points. Let's pick a few points: (-3, -1), (-2, 0), (-1, 1), (0, 1.2), (0, 2.3), (1, 2.5). The change in y from x=-3 to x=-2 is 1 (0 - (-1)). The change in y from x=-2 to x=-1 is also 1 (1 - 0). The change in y from x=-1 to x=0 is not 1 because there are multiple y values for $x=0$. Since the rate of change is not constant, the function is not linear. The points cannot be connected by a line because the graph is discrete.
As x increases by 1, y increases by a constant amount (approximately 1) for so it is linear for this section. When , however, this is no longer true, since there are multiple values for . So, in general, the graph is not a linear function.
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7. The domain of a function is the set of all possible input values (x-values). From the graph, the x-values are -3, -2, -1, 0,
1. Therefore, the domain is $\{-3, -2, -1, 0, 1\}$.
The range of a function is the set of all possible output values (y-values). From the graph, the y-values are -1, 0, 1, 1.2, 2.3, 2.
5. Therefore, the range is $\{-1, 0, 1, 1.2, 2.3, 2.5\}$.
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6. The y-values increase by approximately 1 amount for every unit increase in x-values. Also, the points on the graph cannot be connected by a line. The function shown is not a linear function.
3. Final Answer
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7. Domain: $\{-3, -2, -1, 0, 1\}$
Range:
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