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Q&A #6118


Domain and range of linear functions

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From: Ron (for Teacher2Teacher Service)
Date: Apr 12, 2001 at 06:54:13
Subject: Re: Domain and range of linear functions

Hi Charlene:

Domain and range are extremely important in the study of mathematics.
Initially, they seem so obvious that one wonders what the fuss is about but in
the later grades a thorough understanding will help in the understanding of
many concepts.

I am working under the assumption that you are inquiring about domain and
ranges in linear functions.

First a few definitions:
1. A linear function is a function of the first degree. If y is a linear
function of x, then y is related to x by an equation like y = 3x + 5.
The following are not linear functions of x: tan x (trig function), x^2 - 4
(quadratic function - function of the second degree) {x^2 means x times x or x
squared}.
2. Domain: The set of possible values for which a function exists. For
example: the square root of x in ordinary algebra exists only for non-negative
values of x. The domain of this function is zero and all positive numbers. In
ordinary circumstances the square root of (-5) doesn't exist.
3. Range: The set of all values adopted by a function as x takes on all the
values of the domain. In other words the y values. It can also be thought of
as: the set of values you get as results when you substitute the values in the
domain for the variable.
For example: the range of the square root of x is restricted to all real
numbers equal to or greater than zero.

A more practical example:
If your function is described using ordered pairs such as {(0,-3), (1,2),
(2,7), (3,12)} the domain is {0,1,2,3} and the range is {-3, 2, 7, 12}

Another way to think about a function is as a machine.  Functions are machines
which assign to every INPUT (x) value an OUTPUT (y) value. The machine
operates as follows:
put a value in (say 0) and it produces an output value (-3).
The function machine has just produced an ordered pair (0, -3) for this
particular function! We could, of course, continue forever, putting every
conceivable number in, wait for the machine to produce an output value,
the values we put in (and lead to an answer) are called the domain and the
values that come out are the range.

If you are inquiring about domain and range with regards to non-linear
functions you should look at the following excellent pages from the Ask Dr.
Math Archive

Domain and range
<http://mathforum.org/dr.math/problems/jenni8.25.97.html>
Domain and Range of a Graph
<http://mathforum.org/dr.math/problems/summer10.12.98.html>
Function Machine
<http://mathforum.org/dr.math/problems/johnson10.26.96.html>

I hope that I have given you a clearer picture. If not, please write us again.

Cheers.
 -Ron, for the T2T service

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