http://www.ck12.org Chapter 2. Polynomials and Rational Functions
2.7 Holes in Rational Functions
Here you will start factoring rational expressions that have holes known as removable discontinuities.
In a function likef(x) =(^3 x+(x^1 −)( 1 x)−^1 ), you should note that the factor(x− 1 )clearly cancels leaving only 3x−1. This
appears to be a regular line. What happens to this line atx=1?
Watch This
Watch the first part of this video. Focus on how to identify the holes.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60822http://www.youtube.com/watch?v=OEQnQNvJtG0 James Sousa: Graphing Rational Functions
Guidance
A hole on a graph looks like a hollow circle. It represents the fact that the function approaches the point, but is not
actually defined on that precisexvalue.
The reason why this function is not defined at−^12 is because−^12 is not in the domain of the function.
f(x) = ( 2 x+ 2 )·(x+(^12) )
(x+^12 )
As you can see,f(−^12 )is undefined because it makes the denominator of the rational part of the function zero which
makes the whole function undefined. Also notice that once the factors are canceled/removed then you are left with
a normal function which in this case is 2x+2.