Why Are Functions Important?
Date: 02/03/2005 at 10:52:56 From: Pardue Subject: Functions Why does it matter if something is a function or not?
Date: 02/03/2005 at 22:56:16 From: Doctor Peterson Subject: Re: Functions Hi, Pardue. That depends on the context, of course. I suppose you're asking why we teach how to determine whether a relation is a function or not. For some ideas on why the concept of a function is important, see Why Do We Have Functions? http://mathforum.org/library/drmath/view/62559.html As to why we would want to determine whether something is a function, consider the definition of a function. First, if it is a function, then we can calculate its value given the input; there is only one answer. If it is not a function, then we can't know which answer to choose. That's certainly worth knowing. For example, we know that the square root would not be a function if we allowed it to mean either root, so we choose to define the radical symbol as representing only the principal root. That decision was made essentially so that we would have a function, and expressions involving it would not be ambiguous. We want a symbol to have only one meaning, where possible. It's also worth knowing that when we work with complex numbers, the square root is NOT a function; there is no way to consistently define the square root as having only one value. Knowing that can help us avoid some common mistakes. Similarly, in certain fields of math we define functions in roundabout ways, and have to show that it is "well-defined", meaning that it really has just one value that is not dependent on how we define it. As an example, you might define a function on rational numbers based on the numerator and denominator of the fraction, and you have to make sure that you will get the same answer regardless of which equivalent fraction you use to do the calculation. That is an example of determining whether something is a function, and it amounts to asking, does what I'm doing make sense at all? Further, there are some things we can do only with functions-- differentiating and integrating in calculus come to mind, and it is my understanding that the concept of function was invented primarily so that we could talk about what those operations do (change one function into another). We have to know we have a function in order to do those things. Then, too, the fact that a relation is a function is just one more thing to know about it, which can help in graphing it or otherwise working with it. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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