http://www.ck12.org Chapter 2. Polynomials and Rational Functions
- Identify the holes of the following function.
f(x) =x·sinsinxx
Answers: - First factor everything. Then, identify thexvalues that make the denominator zero and use those values to find
the exact location of the holes.
f(x) =(x+(^2 x)(+ 3 x)(+x^3 +)( 2 x−)^1 )
Holes: (-3, -4); (-2, -3) - The function seems to be a line with a removable discontinuity at (1, -1). The line is has slope 1 andy-intercept
of -2 and so has the equation:
f(x) =x− 2
The removable discontinuity must not allow thexto be 1 which implies that it is of the formxx−−^11. Therefore, the
function is:
f(x) =(x−^2 x−)(x 1 −^1 ) - While the function is not a rational function because it includes a trigonometric expression, the exact same tools
apply. You should ask yourself: when is the sine function equal to zero? Since the sine function is one of the basic
functions you can sketch the function and note that it has a height of 0 at 0,±π,± 2 π...
Since the function is just the linef(x) =xwith holes everywhere the sine function is zero, there are an infinite
number of holes. The holes occur at( 0 , 0 ),(π,π),(−π,−π),( 2 π, 2 π)...
Practice
- How do you find the holes of a rational function?
- What’s the difference between a hole and a removable discontinuity?