2.7. Holes in Rational Functions http://www.ck12.org
This is the essence of dealing with holes in rational functions. You should cancel what you can and graph the
function like normal making sure to note whatxvalues make the function undefined. Once the function is graphed
without holes go back and insert the hollow circles indicating whatxvalues are removed from the domain. This is
why holes are called removable discontinuities.
Example A
Graph the following rational function and identify any removable discontinuities.
f(x) =(^13 (x+ 2 )^2 − 4 )·((xx−− 11 ))
Solution: This function is already separated into two parts, the rational part and a parabola. To graph the function,
simply graph the parabola and then insert a hollow circle at the appropriate height atx=1.
The hole is at (1, -1) because after removing the factors that cancel,f( 1 ) =−1.
Note that most problems will require significant algebraic steps to reach this point. This example emphasizes that
the backbone of the function is essentially a parabola with only one difference.
Example B
Graph the following rational function and identify any removable discontinuities.
f(x) =−x^3 +^3 x−x^21 +^2 x−^4
Solution: This function requires some algebra to change it so that the removable factors become obvious. You
should suspect that(x− 1 )is a factor of the numerator and try polynomial or synthetic division to factor. When you
do, the function becomes:
f(x) =(−x^2 +(^2 xx−+ 14 ))(x−^1 )