SWBAT determine the domain and find the hole(s) of rational functions.
(Section 3-4)
Warm up:
Find the domain for each function:
a) F(x) =
b) R(x) =
c) F(x) =
2x 2  4
x5
1
x2  4
 x2  2
3
x2 1
d) F(x) =
x 1
1
Ratios of integers are called rational numbers. Similarly, ratios of
polynomial functions are called rational functions.
Definition: A rational function is a function of the form:
R(x) =
p( x)
q( x)
Where p and q are polynomial functions and q is not the zero
polynomial. The domain consists of all real numbers except those
for which the denominator q is 0.
In general, the domain of a rational function of x includes all numbers except x-values
that make the denominator zero. Much of the discussion of rational functions will
focus on their graphical behavior near the x-values excluded from the domain.
One major difference between polynomials and rational functions is that a polynomial
is defined for every real number, whereas a rational function is defined everywhere
except where its denominator is zero.
An important question when analyzing the behavior of a rational function is:
What impact does a zero in the denominator of a rational function have on the
shape of its graph?
This question has two answers: The graph can have a hole - that is a place
where a single point is missing – or it can have a vertical asymptote - that is a
place near which the output values explode.
2
Locating Holes of Graphs of Rational Functions
Example 1) Examine the behavior of a rational function near a hole.
r ( x) 
x2  2x  3
x3
a) Give the domain of r.
b) Examine the limiting behavior of r as x approaches 3 from either side.
(1) Find values of r(x) as x approaches 3 from the left - that is, as x gets closer and closer to 3 but
is always smaller than 3 - by using your calculator to fill in the following table (except for the
box in the last row).
Calculate the values of r(x) to at least four decimal places.
x
2.0
2.5
2.75
2.9
2.99
2.999
2.9999
r(x)
2) Find values of r(x) as x approaches 3 from the right - that is, as x gets closer and closer to 3
but is always greater than 3 - by using your calculator to fill in the following table (except for
the box in the last row).
Calculate the values of r(x) to at least four decimal places.
x
4.0
3.5
3.25
3.1
3.01
3.001
3.0001
r(x)
3
Summary
4
Homework:
For each problem find the following
a) Give the domain of the function.
b) Express the function in reduced form, along with its (original) domain.
c) Find the (x, y) – coordinates of any holes in the graph of the functions.
t2
1) f (t )  3
t  9t
2s 2  2
2) h( s )  5
3s  3s 3
x5  x4  x3
3) q ( x ) 
x2  1
5