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Domain and Range of a Function

Date: 03/05/2004 at 08:00:58
From: Veronica
Subject: how do you do domain and range in algebra

Given f(x) = 2x, find the domain and range of the function.



Date: 03/06/2004 at 00:25:39
From: Doctor Achilles
Subject: Re: how do you do domain and range in algebra

Hi Veronica,

Thanks for writing to Dr. Math.

I like to think of functions as little machines (I know, it sounds 
really silly, but it's MUCH more intuitive than thinking about 
abstract math terms).  A function is a machine that takes a number, 
does something to it, and then hands you a number.

[There is one other rule, a function must always give you the same 
output every time you give it a given input, but that's not very 
important for this discussion.]

The simplest function I can think of is:

  g(x) = x

This little machine takes any number you give it, does nothing at all 
to the number, and then hands it back to you.  If I give it 6, it will 
give me 6.

Another pretty simple function is:

  h(x) = x + 1

This machine takes any number you give it, adds one to it, and then 
gives you the result.  If I give it 6, it will give me 7.

What is "domain"?

The domain of a function is a list of all the inputs that the little 
machine can take.  So what does THAT mean?

Some little machines are pretty much indestructible, but other 
machines will break if you give them a certain input.  For example, 
the little machine g(x) = x is extremely tough.  It doesn't care what 
you give it, it doesn't try to do anything at all to it, it just 
hands it right back.  The domain of this little machine is 
all numbers.  A more formal way of writing that is x = {all real numbers}.

The little machine h(x) = x + 1 is also pretty tough.  Because it is
always possible to add 1 to any number, you can give this little
machine any number you want and it will be able to give you back that
number plus one.  The domain of this little machine is also all numbers.

But let's look at another little machine:

  i(x) = 1/x

This little machine takes any number you give it and does the 
reciprocal.  So if you give it 6, it will hand you back 1/6.  If you 
give it 1/2, it will hand you back 2.  This little machine is kind 
of tough.  For almost any number you give it, it will have no trouble 
handing you back the reciprocal.  But this little machine has a 
weakness.  If you give it the number 0, it will break, because 1/0 is 
undefined.  So the domain of this little machine is all the numbers 
except 0.

What is the domain of your function?

  f(x) = 2x

This little machine takes any number you give it and multiplies by 
2.  Is there any number that will break this machine?

What about range?

So we've defined the "domain" of a little machine as all the inputs 
it can possibly take without breaking.  Now let's define the "range" 
as all the OUTPUTS it can possibly give.

Let's think about the range of:

  h(x) = x + 1

Well, for any number I wanted the little machine to give me, I can 
find a number that is one smaller than it and put it in.  So the 
machine can output any number at all.  So its range is all numbers.

Now let's look at

  i(x) = 1/x  

Well, I can get out any whole number I want by putting in a fraction.
So I can get 7 by putting in 1/7, or I can get 1549 by putting in 
1/1549.  I can also get out any fraction I like.  I can get 1/8 by
putting in 8, or I can get 3/10 by putting in 10/3.  I can also get
any negative number I want using the same ideas, but just using a
negative input.  However, there is no number I can put in that will
give me 0 as an output.  So the range of this little machine is all
the positive numbers AND all the negative numbers (but not zero).

Let's look at one more example:

  j(x) = x^2

["x^2" is supposed to mean "x-squared", that is, "x*x"]

First, since we haven't looked at it yet, what is the domain of this 
little machine?  The domain is any number, right?  There is no number 
you can put in that will break the machine.

What is its range?  Well, you can get 0 by putting in 0.  You can get 
any positive number two different ways: you can put in its positive 
square root or its negative square root.  For example, putting in 3 or 
-3 will both cause the machine to put out positive 9.

But there is no way to get a negative number out of this machine.  No
matter what input you give, the answer will be positive.  So even
though its domain is all numbers, its range is only all the positive
numbers AND zero (but not the negatives).

What is the range of your little machine: f(x) = 2x?  Is there any 
number that you cannot get it to output?

Hope this helps.  If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Functions

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