Why Do We Have Functions?
Date: 04/01/2003 at 11:30:00 From: Lisa Subject: Functions Why do we have functions? For example, f(x) = x^2, then f(2) = 2^2 = 4. Or why do we have things like g(x) = x + 3, find g[f(2)] ?
Date: 04/01/2003 at 12:42:43 From: Doctor Peterson Subject: Re: Functions Hi, Lisa. Basically, the concept of functions gives us a way to name the whole process of evaluating a particular expression, so we can talk about it as a whole. We can compare different functions, discuss their properties, or actually operate on functions to make new functions. It also broadens the concept, because not all functions can be written as a simple expression. These two processes, naming things and extending them, are central to what mathematics is all about. For example, the first function you showed can be called 'squaring', and the second can be called 'adding 3'; but most functions would have to have much more complicated names. By calling one F and the other G, we have a simple way to discuss them. Some functions, like the square root and the absolute value, can't be expressed in terms of more basic functions, but only by inventing a whole new symbol. In fact, we like to write the square root as 'sqrt(x)', using function notation, because we don't have the symbol available in e-mail. We can also treat these names like variables, where we don't know what specific functions we are calling f and g, yet we can say general things about the relation of f and g, proving that something is true for ANY functions, or at least for any functions of a certain type, all at once. That is powerful! Composition of functions takes two functions and makes a new one out of them. The inverse of a function is a new function that has some important properties; in fact, if you think of composition as a sort of "multiplication" of two objects (which are whole functions), then the inverse function is sort of a reciprocal. In fact, that's why we use the notation we do for inverse functions. We've taken a familiar idea from arithmetic and applied it to something far bigger, largely just by having named functions. Composition can be thought of as something like plumbing or electrical wiring, where we buy parts off the shelf and connect them end-to-end to make a new device or to wire a house or factory according to our needs. We know how each pipe, wire, switch, etc. functions (pun intended), and we know how they combine, so we can understand the whole complicated system in terms of its parts. Without the function concept, we couldn't do that in math. The concept is most useful when you get to calculus, and find that the derivative and the integral are operations on functions: given one function f, you can make a new function f' out of it, that has certain important properties. This moves math up one level from algebra. So the concept of functions is essential for a good understanding of calculus. The concept of a function is also central to computer programming, though the details are somewhat different there. Most of what a programmer writes consists of 'functions' that do parts of the work of the program. By designing functions that do little pieces, we can string them together to do more complicated things without looking so complicated. For example, the sqrt function I mentioned gets its name from many computer programming languages that provide this and many other built-in functions so we don't have to write them ourselves. By making that just part of a more general concept of functions, we are able to write our own functions, and then put those together to make larger functions. Here are several explanations of various aspects of these ideas: Why use f(x)? http://mathforum.org/library/drmath/view/54577.html Are All Functions Equations? http://mathforum.org/library/drmath/view/53273.html Inverse Functions in Real Life http://mathforum.org/library/drmath/view/54605.html 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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