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/