Square Root Function

Date: 12/16/98 at 18:37:26
From: Jason Lynch
Subject: Square root problem

Given this problem: 

   sqrt(2x + 3) = -1

solve for x.

I say the answer would be -1, because:

   sqrt(2(-1) + 3) = -1
      sqrt(-2 + 3) = -1
           sqrt(1) = -1

My second year Algebra teacher says otherwise. She says that there 
would be no solution. I fail to see the reasoning, as -1 is one 
possible square root of 1.

Jason Lynch

Date: 12/17/98 at 16:58:23
From: Doctor Rick
Subject: Re: Square root problem

Hi, Jason! You've asked a good question.

What's going on here, I think, is that we have two separate concepts 
that can easily be confused. 

One concept is the process of finding the root of a square; for 
instance, the number x such that x^2 = 4. You know that this has two 
solutions: 2 and -2 are both roots of this equation.

The other concept is the square root FUNCTION. A function takes in one
number and returns another number. The number that it returns must be 
UNIQUE since a function is by definition single-valued. You can't put 
in 4 and get out both 2 and -2.

The square root function is therefore DEFINED so that sqrt(x) returns 
the NON-NEGATIVE root of y^2 = x. Then we say that the two roots of 
x^2 = 4 are +-sqrt(x), plus or minus the [non-negative] square root 
of x.

Admittedly this is a rather arbitrary definition, and your teacher's 
answer feels artificial. But if functions were not single-valued, we'd 
have a mess. And when you use the square root function in real 
situations, it will make sense. For instance, the distance between the 
points (0, 0) and (x, y) is sqrt(x^2 + y^2), and it makes sense that 
the distance is a non-negative number.

Here is something from our Archives that might add a bit to my 
reasoning.

   http://mathforum.org/dr.math/problems/ken.8.28.96.html   

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