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Topics: Rational function, Function, Limit of a function Pages: 33 (573 words) Published: October 13, 2014
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) 

3  2x  x2
x2  x  2

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

3  2x  x2
x2  x  2

Horizontal Asymptotes

Degree of numerator = Degree of denominator
Horizontal Asymptote:

coefficient of denominator

Degree of numerator < Degree of denominator
Horizontal Asymptote:

y

coefficient of numerator

y0

Degree of numerator > Degree of denominator
Horizontal Asymptote:

none

Find all asymptotes & holes & then graph:
f ( x) 

x2
2x  3

Find all asymptotes & holes & then graph:
f ( x) 

x2
2x  3

Find all asymptotes & holes & then graph:
f ( x) 

x 1
x  2x  3
2

Find all asymptotes & holes & then graph:
f ( x) 

x 1
x  2x  3
2

Find all asymptotes & holes & then graph:
f ( x) 

x2  3x  4
x4

Find all asymptotes & holes & then graph:
f ( x) 

x2  3x  4
x4

Find all asymptotes & holes & then graph:
f ( x) 

3x2  7 x  6
x2  9

Find all asymptotes & holes & then graph:
f ( x) 

3x2  7 x  6
x2  9

Find all asymptotes & holes & then graph:
f ( x) 

 x  3
x2

2

Find all asymptotes & holes & then graph:
f ( x) 

 x  3
x2

2

Find all asymptotes & holes & then graph:
f ( x) 

3x2
2x2  5

Find all asymptotes & holes & then graph:
f ( x) 

3x2
2x2  5

Find all asymptotes & holes & then graph:
f ( x) 

x
x2  2x  3

Find all asymptotes & holes & then graph:
f ( x) 

x
x  2x  3
2