Finding the Domain of a Function

Date: 9/28/95 at 15:29:13
From: Anonymous
Subject: calculus

What is the domain of f(x)=((2-x)/(x-2))^.5 ?

Date: 9/28/95 at 17:44:34
From: Doctor Jonathan
Subject: Re: calculus

Since the square root is only defined to be real over non-negative 
numbers, the domain of this function is the solution to the inequality:

(2-x)/(x-2) >= 0

This will occur when the numerator and the denominator have the 
same sign, or when the numerator is 0.  Well, the numerator in this 
case is only positive for x > 2, and the denominator is only positive 
for x < 2.  In other words, they have different signs (they are in fact 
opposite) over all x.  

So what happens at x = 2? Well, here the 
denominator ruins things for us by being zero, and so the function is 
undefined even at x = 2.  So, there are no values of x such that f(x) is 
defined.  As such, the domain is the empty set.

Hope this helps...

-Doctor Jonathan,  The Geometry Forum