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I am in need of dier assistance if i want to pass

  1. Sep 11, 2005 #1
    ok i am in serious trouble in my math class alos in spelling some of u might think of my word dier i don't think thats how you spell it but whatever. as i was saying i am totally lost in my class not to mention my homework :cry:. Could someone just define function, domain, range, and how to figure all of these out. I also have a major problem with piece wise functions. I see no use for them expect to make me go bald from ecessively pulling my hair out :biggrin: . anway if u were wondering i'm in analysis and i have a quiz tom which i will probably fail. i've been studying but i decided that i cannot do this alone. i know i'm really acting like a drama queen but i just can't figure this stuff out. so any help would be really really great. thanks. :smile:
     
  2. jcsd
  3. Sep 11, 2005 #2

    matt grime

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    Dire, as in dire straits.

    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.
     
  4. Sep 11, 2005 #3
    You are in Grade 12 Calculus (and Analysis)? The rest of my post will assume that. I suggest making friends with some people in your class that are good at math and also talking to the teacher to look for some assistance. Ditch your pride and start asking questions in class as well and ignore any snickers from your classmates.

    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.
     
  5. Sep 11, 2005 #4

    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.

    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 exactly one element in the range. Look at the relation [itex]y=\pm \sqrt{x}[/itex]. Pick the number 4. Now clearly 2 satisfies this relation, but so does -2. Which one do you pick? This relation is not a function. A function is an exact rule and must tell you exactly which value in the range to pick for a given value in the domain. Here is a quick way to tell if a relation is a function by looking at the graph: The way you tell what value of y to pick for a given value of x is to look right above the value of x and see at what value of y it touches the graph. If you see that, when you look above any value of x in a vertical line, this line touches the graph more than once, you know the graph is not of a function. This is because it just gave you two values of y to choose from for a given value of x, which we said functions can not do.
     
    Last edited: Sep 11, 2005
  6. Sep 11, 2005 #5

    matt grime

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    BIG nitpick that ought to be ignored and is aimed at the cogniscenti, as a warning, and for you to bear in mind for later:

    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.
     
  7. Sep 11, 2005 #6
    As far as piecewise functions go, they play a very important role in mathematical expressions of physical phenomena (not sure if phenomena is the correct word here). For example, a circuit is turned off from time = 0 until time t0. At t0 the circuit has an equation of 2t+3. You would then be able to express it in the equation

    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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