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Question 1203313: Decide whether the relation defines y as a function of x. Give the domain and range.
x+y<3
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
A relation is a function if for every value of x in the domain there is only one y value.
For a linear inequality like this, that is never the case; for any x value there are an infinite number of y values.
For a very simple example, (x,y) = (0,1) and (0,2) both satisfy the inequality.
As for the domain and range, it should be easy to see that both x and y can be any real numbers.
Note I can't imagine the concept of function ever being applicable to an inequality. With an inequality, there will always be an infinite number of y values for any x value.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
The relation x+y < 3 is not a function because an input like x = 2 corresponds to infinitely many y outputs.
x+y < 3
2+y < 3
y < 3-2
y < 1
Simply select any value smaller than 1
Therefore points like (2,0), (2,-1), (2, -2), etc are all solutions.
They form a vertical column.
As such, this example is a visual indication we have failed the vertical line test.
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If on the other hand the relation is x+y = 3, then this would be a function. Each x input corresponds to one and exactly one y output. The graph of x+y = 3, aka y = -x+3, passes the vertical line test.
This diagonal line passes through (0,3) and (3,0). These points represent the y-intercept and x-intercept respectively.
The green graph passes the vertical line test. This is because it is impossible to draw a single vertical line through more than one point on the green line.
With regard to the function y = -x+3, we have:
Domain = set of all real numbers
Range = set of all real numbers
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