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- Thread starter Analysishater101
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matt grime

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A function between two sets A and B is a way to associate to each element in A exactly one element in B. It can be thought of as a box that takes in ANY element in A and spews out a unique element in B.

Eg, the functionf(x)=x^2 from the natural numbers to the natural numbers. IT takes n and spits out n^2.

the domain is the set A that it takes in, and the range is the subset of B that it spits out. Note in the previous example the domain is the natural numbers, but the range is only the subset of the natural numbers that are perfect squares.

You are probably used to functions that are defined nicely, like sin(x) or x^2, but not al functions are like that, and indeed the most useful one in physics and engineering is proabably defined piecewise like this:

H(x)=0 for x<1 and 1 for x=>1

the heaviside fucntion.

notice we split the domain into different parts (ooh, let's call them, I don't know, pieces, perhaps?) and we define it on each piece. Let's say we are defining it "piecewise" in the natural labnguage sense.

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A function is basically a graph that passes the "vertical line test" which is a fancy way for saying for every x value there is exactly one y value. e.g. y = x is a function because it's just a straight line and for every x value there's one y value. a circle isn't a function because for some x values there are 2 y values. Picture it.

A domain is simply what values x can be allowed. For instance; for the function y = root(x), x can't be negative because that would give you an undefined number. The domain for that would be x is any element of the reals, given x is greater or equal to zero.

A range is simply what values y can be allowed; same thing.y = root(x), it's obvious that if you look at the graph that y can't have any negative values. So the range would be y is any element of the real numbers, given y is greater or equal to zero.

A piecewise function is simply a set of rules that govern a function. Treat it as "laws" that manipulate how a function will work.

Please consult your textbook or such because these are very simple concepts that you should be able to do with complete confidence in order to move on in mathematics.

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LeonhardEuler

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Alright, I will try to explain these definitions, but you will have to keep in mind that these definitions are not standard; there are slight differences between the definitions of functions and ranges in different textbooks.Analysishater101 said:

What is a function? A function is simply a rule that takes a number from one set of numbers, called the domain, and returns exactly one number from another set of numbers, called the range. So "doubling" is a function. You give it a number, say 4, and it uses some rule to give a number in return, 8 in this example. The domain in this case is all real numbers. This is because if you give any real number, the function will always be able to give a number in return. The range is also all real numbers. This is because if you pick any number, say 20, then there will always be a number that will give you 20 when you put it into the function(10). If you denote elements in the range by "y" and elements in the domain by "x", then the rule for the "doubling function" can be written as y=2x.

Now consider the "squaring function", [itex]y=x^2[/itex]. In this case the domain is again all real numbers because, for any real number, there is exactly one number that is its square. But think about the range. 16 is in the range because there is a number, 4, that will give you 16 when you put it in the function. But look at -16. There is no number in the domain that will give you -16 when you square it. The range is all non-negative real numbers because only non-negative numbers come out of the function [itex]y=x^2[/itex].

It is important to note that each element of the domain must give

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matt grime

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there is NO debate abuot the definitions of anything, they are completely standard. BUT most calculus textbooks (in the american sense) are wrong in their definitions. Indeed the questions like: determine the domain of sqrt(x^2-1) are complete crap since the domain is part of the required information for anything to be a function.

The moral of the story? Remember that the books are written by people who make mathematicians' blood turn cold and that:

inputs and outputs shold be assumed to lie in the real numbers, and that when asked about "find the maximal domain" that they mean find all real numbers we can put in the function so that we never divide by zero or take a square root of a negative number.

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f(t) = {0, 0<t<t0

2t+3, t>=t0}

This also works for the dirac-delta function, which expresses an impluse such as a hammer blow.

I found that knowing the uses of the mathematics one is learning greatly increases the persons desire to learn it. I know when I was in high school I could not stand math because no one told me how I could apply it.

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