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One Element of Domain Taking on Two Y values
- Thread starter Jason76
- Start date
An x value can take two y values, but a y value cannot take on two x values. Why?
If you are trying to determine if a relation is a "function", you have it backwards. Each x can have at most one y value. Look here for a description of the "vertical line test.":
http://www.youtube.com/watch?v=-xvD-n4FOJQ
I like to think of univariate function y = f(x) as a RULE for taking an input or argument, x in this case, and transforming it UNAMBIGUOUSLY into an output or result, y in this case. Another way that this is said is that x is the independent variable. But the idea is that once you know x you also know y WITH CERTAINTY.An x value can take two y values, but a y value cannot take on two x values. Why?
So if f(2) = 3 and f(2) = -107, what does f(2) mean? There is an ambiguity.
However, f(2) = 4 and f(-2) = 4, there is no uncertainty: each value of x takes you to a definite value of y. There is no reverse inference that each value of y takes you to a definite value of x. The dependent variable is determined by the independent variable, but the dependent variable does not determine the independent one. Each child has a unique biological father, but not every biological father has a unique child.
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
Y taking on two x values makes sense: Because by looking at a parabola \(\displaystyle y = x^{2}\), you will find that at some point y = 2 which shares itself with two x values -2 and 2.
More generally, if you go into a store and see two different items that happen to have the same price, that would not be unusual. If you saw two of the same item with different prices, you would think one had been marked wrong.
If you were to look over the grade lists for a school and see two different students in two different classes, with the same grade, that wouldn't bother you. If you see the same student, in the same class, with two different grades, that would bother you.
One of the requirements of a good scientific experiment is "repeatability". If different scientists, on different days, do exactly the same experiment, they should get (to within a reasonabe "error range") the same result.
It is considerations like those that lead to the non-symmetric definition of "function", that if \(\displaystyle x_1= x_2\), we must have \(\displaystyle f(x_1)= f(x_2)\) but it might happen that different values of x give the same y.