what competitive forced have challrnged the movie industry?

Rational Expressions,
Asymptotes, and Holes
Dr. Litan Kumar Saha
Assistant Professor
Department of Mathematics
University of Dhaka

Rational Expression


It is the quotient of two polynomials.



A rational function is a function defined by a rational expression.

Examples: y x2 x5 y

3x2  2 x  5 x3  4 x2  5 x  7

Not Rational: y 4x x2 y

x x2  5

Find the domain:
Graph it:

f ( x) 

1 x Find the domain:
Graph it:

f ( x) 

1 x Find the domain:

Graph it:

f ( x) 

1 x2 Find the domain:

Graph it:

f ( x) 

1 x2 Vertical Asymptote


If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.

Find the domain:

Graph it using the graphing calculator.
What do you see?

f ( x) 

x3 x  x  12
2

Find the domain:

Graph it using the graphing calculator.
What do you see?

f ( x) 

x3 x  x  12
2

Hole (in the graph)


If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote.



The exact point of the hole can be found by plugging b into the function after it has been simplified. Find the domain and identify vertical asymptotes & holes. f ( x) 

x 1 x2  2x  3

Find the domain and identify vertical asymptotes & holes. f ( x) 

x 1 x2  2x  3

Find the domain and identify vertical asymptotes & holes. f ( x) 

x x2  4

Find the domain and identify vertical asymptotes & holes. f ( x) 

x x2  4

Find the domain and identify vertical asymptotes & holes. f ( x) 

x5
2x2  x  3

Find the domain and identify vertical asymptotes & holes. f ( x) 

x5
2x2  x  3

Find the domain and identify vertical asymptotes & holes. f ( x) 