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Finding the Domain of a Function

Date: 02/10/2009 at 11:14:05
From: Brandon
Subject: Domain of a function

I'm having trouble finding the domain of a function.  I'm trying to do
all of this on an online math class and its just extremely hard to
understand all of it by reading it off a web page instead of having it
taught in front of me.



Date: 02/10/2009 at 14:19:00
From: Doctor Ian
Subject: Re: Domain of a function

Hi Brandon,

One thing that works for me is to actually read my textbook (or a web
page, or any other written resource) out loud, word by word... as if
someone else is reading it to me.  You'd be surprised at how much this
can help you, if only because it keeps you from skimming past a few
words here and there that turn out to be critically important. 

(By the way, if you're planning to go to college, learning to teach
yourself from written resources is a skill that you need to be
polishing.  When you're in college, classes aren't where the teacher
reads the book to you, as in high school.  Classes are where the
professor discusses material that he assumes you've already read
before class.)

Anyway, the domain of a function is the set of legal inputs.  Given a
function like

  f(x) = 2x

we can put in any real number for x, and get a sensible value for the
function.  So the domain is "all real numbers".  Contrast that with 
           __
  f(x) = \/ x

Now we get nonsense if we put in a negative value of x, but we're okay
with any other real numbers.  So the domain is "all non-negative real
numbers" or "all real numbers greater than or equal to 0".

We can change that slightly, so it looks like
           ______
  f(x) = \/ x - 3

and now it works as long as x is not less than 3.  So the domain
becomes "all real numbers not less than 3" or "all real numbers 
greater than or equal to 3". 

Dividing by zero is another no-no, so we might write a function as
                  ______
  f(x) = 1  /  \/ x - 3

Now we can't use 3, either, because it would require us to divide by
zero.  So the domain becomes "all real numbers greater than 3".

What about something like

  f(x) = 1 / [(x + 1)(x - 2)] ?

Now we're fine except if x is -1, or 2.  So the domain is "all real
numbers except -1 or 2".

And so on, and so on. 

Basically, the idea is that you look at your function, and you
consider what values might cause it to "break", by requiring you to
take the square root of a negative number, or divide by zero, or get a
value of sin(x) greater than 1, or anything else that doesn't make
sense.  And you remove those values from the set of real numbers. 
What remains, is your domain.

Does that make sense?  If not, try reading it out loud.  And if it
still doesn't make sense, write back and let me know which sentence
starts to lose you, okay? 

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



Date: 02/10/2009 at 18:51:30
From: Brandon
Subject: Thank you (Domain of a function)

Thank you for the help!  I understand better now.  You were very
thorough and easy to understand, which is a big help.  Thanks again.
And I may be back!
Associated Topics:
High School Functions

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