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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

Functions and Equations
http://mathforum.org/library/drmath/view/53236.html

Are All Functions Equations?
http://mathforum.org/library/drmath/view/53273.html

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

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?
http://mathforum.org/library/drmath/view/62559.html

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

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Functions
High School Polynomials

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