Help Needed: Explaining Functions and Solving a Question

[udaibothra]

Hello Everyone,

I'm 16, and am trying to teach myself Calculus. Its just been a day since I started, and I seemed to be going on well pretty well by myself till I came across the topic of functions.
I do not have access to a math teacher at the moment because my holidays are going on. I was wondering if anyone of you could help me with this.

Can you clearly explain what a function means, what is its use, and some examples of some sums.

Also, while going through the examples in my textbook, I came across a question which asks-

- Does the relation {(x,y) Iy=IxI, x is a rational number)}define an equation? If yes, justify, write the range and draw the graph.

(First of all, the 'I' in the question is not exactly an 'i' but it is a line bracket. I do not know how to make it on the computer. Secondly, what does a line bracket mean? That part, "Iy=IxI" -- what exactly does it mean?

I'm sorry for such a question, but I couldn't find a better place for such a question of mine. I hope you people help.

Thank You. :)

 

[Tedjn]

A function is a black box that takes elements of one set to elements of another set. For example, a function f might take elements in a set X to elements in a set Y. The set X is called the domain of f. The set Y is called the codomain of f. For every element x in X, the function associates an element f(x) in Y. The range (in your context), probably refers to all y in Y such that there exists x in X with y = f(x). In other words, the range is everything in y that is hit by some element in X through f. This is sometimes also called the image. (And to confuse you even more, in some books, range is instead used as a synonym for codomain.)

Depending on what X and Y are, this function f can take on different forms. One common way to show how general the idea of a function is is to take the sets X and Y to be completely unrelated to math. For instance, let X be the set of countries in the world and Y to be the set of all cities in the world. We can let our function f assign to every country in X its capital city in Y. That is, if x = USA, then f(x) = Washington, D.C.

In this case, not every element in Y is assigned to a country in X by the function f. However, every country in X (the domain) must be associated with a city Y in the codomain. That is, f(x) must exist for all x in X. Observe that our choice of the set Y is a little arbitrary. We could have let Y' be the smaller set of all capital cities in the world and have the same function, except from X to Y'. In this case, by our choice of Y', it should be clear that every y' in Y' is associated to a city x in X. I hope that this small bit clarifies the notions involved when talking about a function.

Of course, many function you encounter in math will involve real numbers. Typical functions you see in calculus will be given by an equation. If I say that f(x) = x² + 3, for example, the domain and codomain are usually both understood to be R, the set of real numbers. To every real number x, I associate another real number f(x).Now, is your second question actually: Does the relation \{(x,y)\,|\,y = |x|, x\text{ is a rational number}\} define a function?

If so, do you understand what it means for a relation to be a subset of all ordered pairs (a,b) where a and b are both real numbers? You can read this as, this relation R is a set of ordered pairs. In R are all ordered pairs (x,y) such that the second coordinate y is equal to the absolute value of the first coordinate x, and x is a rational number.

A function can also be treated as a collection of ordered pairs, namely the ordered pairs (x,f(x)) for every x in X. In particular, note that every x appears as the first coordinate in one and exactly one ordered pair. Because, if x appears in two ordered pairs, we are confused what f(x) should actually be (the second coord in the first pair or the second?). If x doesn't appear, then f is not defined for every x in X.

However, your question is a little unclear. I treated the relation as a subset of ordered pairs where both coordinates are numbers in R. But it might well be that your question wants your to consider a subset of ordered pairs where both coordinates are numbers in Q, the rational numbers. If so, that changes the answer.

I know that this post is rather lengthy. I've tried to be as comprehensive as possible without confusing you too much. Try to digest all this, and ask further if there is anything you don't understand.

 

[udaibothra]

Hi Tedjin!

Thank you for such a good explanation! You actually explained it better than our course book!
So am I basically to assume that functions ie.f(x) are present to assign a value to the y variable in the Y set? And in the example you gave me about the countries and the capitals, is it more like- f(capitals) ? I know this doesn't make much sense, but I'm at a loss to put it in a better way.

Can you also tell me a bit about 'ordered pairs?'. And in the numerical equation you gave me-

f(x)= x^2 + 3

What is the domain and the range? And what exactly is the purpose of the word function? Does it mean that we have to find a value for the variable x which satisfies the equation?

I'm sorry for taking up your time, but the way you explained it the previous times provokes me to ask for more.

Thank You Tejdin. :)

(ps- if you have other important things to do, don't waste your time trying to explain these questions of mine. I shall look elsewhere on the internet then! I'm sure I'll find something! :) )

Thnks once again! :)

 

[udaibothra]

Hi Tejdin,

Yes, I do know what ordered pairs are now.

As far as the question is concerned, I'll try to tackle it myself! :)

 

[Char. Limit]

Can you also tell me a bit about 'ordered pairs?'. And in the numerical equation you gave me-

f(x)= x^2 + 3

What is the domain and the range? And what exactly is the purpose of the word function? Does it mean that we have to find a value for the variable x which satisfies the equation?

First, ordered pairs are simply a way to define points on a plane. The x-axis is usually the horizontal, and the y-axis the vertical. Sometimes the t-axis is horizontal. The ordered pairs of a function simply show the value of the function at a point. For example, in your above function, when x=0, then f(x)=3, so I could write (0,3). For other points, the same manner is used, i.e. (1,4) or (-2,7). In general, an ordered pair is usually of the form (x,f(x)) for functions. At least I think so.

A function is supposed to define one variable in terms of another. Your function above is an example. It defines the variable f(x) in terms of the variable x.

To find the domain, we need to find the set of numbers x where f(x) can't exist. Here, I can throw in any real (or complex) number for x and return a valid number. Thus, the domain is the entire real line. The range is the set of numbers that f(x) can take. Now here, x^2 is always positive, so f(x) will never go below three. So the range is all numbers greater than or equal to 3.

I'm not exactly sure, but I believe that by solving the equation for x, you can then see the allowed areas for f(x). Here, the solution for x is...

x=\sqrt{f(x)-3}

So, since a square root is only defined (on the real numbers) for a positive radicand, we know that f(x)-3 must be greater than or equal to zero, written of course as f(x)-3\geq0 which we can simplify to f(x)\geq3.

I hope that this can help.

 

[Tedjn]

In addition, let me try to clarify a few of your other questions. You can use the notation f(capitals), where here we let the unknown variable capitals stand for some actual state capital. However, it is more typical to use short variable names to stand for the unknowns, e.g. x or t (for time).

The purpose for the word function is simply to indicate a relationship between two sets. In calculus, it will usually be between two sets of real numbers, given in the form a formula, e.g. f(x) = x² + 3, as indicated. This relationship is not completely unrestricted. Every element in the domain must be associated with one and only one an element in the codomain. You should picture a function as a black box. For a much more general relationship that does not have these restrictions, we have a relation.

 

[udaibothra]

Thanks Char Limit and Tejdin!

Your explanations really helped a lot! It is almost as if I can understand the sums I come across like words I know the meaning of! I shall come back if I stumble across any more questions which is VERY LIKELY. Thanks once again! Cheers. :)

PS- I'll be traveling for till the 22nd, so I doubt I'll get to use the internet. :)

So don't expect any problems soon.