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How Are Functions and Expressions Related?

Date: 07/09/2004 at 11:05:49
From: Kristy
Subject: relationships between functions and expressions

What is the relationship between a function and an expression?  I
don't see any relationship, they are two completely different things.

Date: 07/09/2004 at 11:42:44
From: Doctor Peterson
Subject: Re: relationships between functions and expressions

Hi, Kristy.

Actually, I would say that they are almost exactly the same thing!  
But that may not be obvious.

An expression is essentially the directions for a calculation, telling 
you to take this variable and that variable and do certain 
multiplications, additions, and so on.  Right?  Here's an example:

  2x - y
  3x + 2y

A function is a "machine" that takes one or more variables and does 
certain things to them to get a number out, which is the value of the 
function.  Here is an example with one variable:

  f(x) = 3x - 4

Here is one with two variables:

           2x - y
  f(x,y) = -------
           3x + 2y

Look familiar?  I can use any expression as the definition of a 
function, since the expression tells what to do with the variable(s) 
to get a value.

On the other hand, not every function can be written as an expression 
using common operations.  Here is a discussion of that fact, and other 
useful ideas about functions:

  Functions and Equations 

See also:

  Are All Functions Equations? 

The terminology all flows together sometimes.  If you were given the 

  2x + 3y = 5

and I wanted you to "solve for y", I might instead tell you to "write 
an expression for y in terms of x" or to "write y as a function of 
x".  These all mean the same thing, but focus on different aspects of 
what you are doing.  "Solving" focuses on the equation as a problem, 
and the particular variable we want to find; "in terms of" focuses on 
which variable(s) are to be used in the calculation, rather than 
which is to be found; and "as a function" focuses on both.  The latter 
is probably preferred by mathematicians for that reason, as well as 
for brevity.  And brevity (the ability to talk about a big concept in 
a single word) is the main reason for defining "functions":

  Why Do We Have Functions? 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Definitions
High School Functions
High School Polynomials

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