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May 10th, 2017, 06:40 AM   #1
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on Vertical line test and definition of a function

Vertical line test
“A function can only have one output, y, for each unique input, x.” – from Wikipedia


Why do we want a function to have only one output?
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May 10th, 2017, 07:25 AM   #2
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by definition a function evaluated at a point must have only one result.

function mappings cannot be one to many
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May 10th, 2017, 09:06 AM   #3
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Quote:
Originally Posted by levisitor View Post
Why do we want a function to have only one output?
Imagine you owned a vending machine company. How happy would you be if every time a customer input $1.50 they got out two bottles of water rather than one?

You are sitting on your couch at home. You grab the remote control. You input channel 49. How happy would you be if you got ESPN and PBS simultaneously broadcast on top of one another?

If these things happened, then your vending machine and your remote control would not be FUNCTIONing properly.
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May 12th, 2017, 04:06 AM   #4
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This is related to the basic science idea of "repeatable" results- if you do the same experiment with exactly the same conditions, you should get the same result. That is the result of the experiment is a function of the initial conditions.
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May 12th, 2017, 04:58 AM   #5
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Originally Posted by levisitor View Post
Why do we want a function to have only one output?
For me, there are two main reasons:
  1. It makes the analysis easier. If we want mutiple-valued functions, we can create them from single-valued functions. For example: a circle of radius $r$ centred on the origin can be written as $x^2+y^2=r^2$, but this doesn't define a function $y=f(x)$. But, if we write $x=r\cos(t)$ and $y=r\sin(t)$, we get the same curve where both $x$ and $y$ are functions of $t$
  2. The value of functions is their ability to predict outcomes. If your "function" gives two (contradictory) results, how is that a prediction? It is essentially saying "I don't know". The scientific solution is to further analyse the problem to identify the conditions that determine which result will occur. These are then additional inputs to a single-valued function that does predict usefully. (Note that the example above does this in some sense).
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May 13th, 2017, 05:05 PM   #6
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Thank you for your great replies, I really appreciate it.
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May 13th, 2017, 09:53 PM   #7
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Imagine you get into bed with your girlfriend. How happy would you be if she brought her smoking hot sister?

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May 13th, 2017, 10:11 PM   #8
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Imagine you get into bed with your girlfriend. How happy would you be if she brought her smoking hot sister?
In complex analysis we call that a Riemann surface!!
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