Functions and Equations

Date: 02/10/2001 at 18:56:31
From: Abby Power
Subject: Functions and Equations

When is an equation also a function, and how do you recognize a 
function?

Date: 02/10/2001 at 20:35:51
From: Doctor Peterson
Subject: Re: Functions and Equations

Hi, Abby.

I wouldn't say an equation is ever a function; but equations can be 
used to define functions.

An equation is any "number sentence" that says two expressions are 
equal.

A function is a relation between two or more variables, such that for 
any value of the independent variable(s), there is exactly one value 
for the function. The function itself can commonly be expressed by a 
name, such as "f"; and it may be defined by stating the function's 
value as equal to an expression:

    f(x) = 2x + 2

Note that the function is f, and f(x) is simply the value of the 
function for a given value of its argument. Nor is the equation the 
function; the equation is being used to tell us the value of the 
function for any x, and thereby to define the function itself.

Not all functions can be expressed this way; for example, the square 
root function can't be expressed other than by the square root symbol. 
We would define that function by saying something like

    sqrt(x) is the non-negative number y for which y^2 = x.

Here the equation y^2 = x doesn't fully define the function by itself.

If you are given an equation relating variables x and y, and are asked 
whether this defines a function of x, you would solve the equation 
for y and see if this can be done in such a way that for every x only 
one value of y will satisfy the original equation. In the case of 
y^2 = x, this can't be done; but by restricting y to non-negative 
values, we can define a function.

If this isn't quite what you wanted, please send me a sample problem 
so I can see what sorts of equations and functions you have in mind.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/