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Square Root Function: Why Restrict Its Range to Non-Negative Numbers?

Date: 01/26/2010 at 01:58:08
From: name
Subject: range of radical function negative or positive

In a problem where you have to find the domain and range of a square 
root function, why is the range non-negative?

For example, in f(x) = sqrt(x - 12), I know that the domain is all 
real numbers greater or equal to 12, but I am having trouble with the 
range part.

My answer key says that the range is all non-negative real numbers, 
but it seems to me that it could be all non-positive real numbers 
instead.  In my opinion, the range of a relation can be all real 
numbers.  In y = sqrt(x - 12), where x is 48, y should work as both 6 
and -6.  After all, (6)^2 = 36, and (-6)^2 = 36, too.  The graph is a 
curve that is symmetrical on the x-axis.

A function has to pass the vertical line test, so one half of the 
graph has to go away. But why not preserve the negative half rather 
than the positive half?  If you only have negative y-values for the 
function, the graph still passes the vertical line test. 

I realize that the solutions for a relation and a function would be 
different, because the function can only have 1 y-value assigned to 
an x-value.  This means that the relation y = sqrt(x - 12) can't be a 
function because its graph has two halves, one above the x axis and 
one below, which fails the vertical line test.



Date: 01/26/2010 at 11:27:18
From: Doctor Peterson
Subject: Re: range of radical function negative or positive

Hi, Name.

Yes, you COULD define a negative-square-root function instead of the 
positive-square-root function that we use; but in this setting, the 
radical symbol is defined to represent the principal root, which is 
defined as the non-negative one.

Now, why DO we prefer the principal root to be non-negative?  Is it 
just an arbitrary choice?

No.  One reason is that it makes the rules for working with a root 
simpler.

You may recognize this rule:

  sqrt(ab) = sqrt(a) * sqrt(b)  for any non-negative numbers a, b

With the sqrt function defined as usual, that rule holds true:

  sqrt(4*9) = sqrt(36) = 6

  sqrt(4)*sqrt(9) = 2 * 3 = 6

But if instead we had a negative (non-positive) root, which I'll call 
nsqrt to avoid confusion, we'd find this:

  nsqrt(4*9) = nsqrt(36) = -6

  nsqrt(4)*nsqrt(9) = -2 * -3 = 6

So the general rule would have to be

  nsqrt(ab) = -nsqrt(a) * nsqrt(b)

That's not as neat, is it?  In math, we generally choose the 
definitions that make things as simple -- and as general -- as 
possible.

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


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



Date: 01/26/2010 at 18:27:12
From: name
Subject: Thank you (range of radical function negative or positive)

Thanks for the fast and helpful answer.
Associated Topics:
High School Functions
High School Square & Cube Roots

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