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Section 5B – Rational Functions
Definition. A rational function is the quotient of two polynomials. That is, if r(x) is
a rational function, then there are two polynomials p(x) and q(x) such that r(x) = p(x)
q(x) .
Facts about rational functions r(x) =
p(x)
q(x) :
• The domain of a rational function is the set of real numbers x such that the
denominator is not 0, i.e., q(x) 6= 0.
• The x-intercepts of a rational function r(x) are the real numbers x, where the
numerator is 0, and the denominator is not zero, i.e., p(x) = 0 and q(x) 6= 0.
Recall, the x-intercepts are also called zeros.
• The y-intercept is the y-value when x = 0, i.e., the y-intercept is r(0). Note
there is at most one y-intercept because r(x) is a function and must pass the
vertical line test. (If 0 is not in the domain of r(x), there is no y-intercept.)
Example 1. Find the domain, x-intercepts, and y-intercept for the function
.
f (x) = 9(x+3)(2x−7)(x+1)
x(4x−1)(x+3)
Domain:
x-intercept(s):
y-intercept:
Horizontal Asymptotes
Example 2. Using the graph, determine the end behavior (the behavior of f (x) as
x → ∞ and x → −∞) for f (x) = x1 .
Example 3. Using the graph, determine the end behavior of the function r(x) =
2x2 −x+5
x2 +1 .
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Generalized Technique for the end behavior of a rational function: To determine the end behavior of a rational function, do the following:
• Find the highest power n of x that is in the denominator.
• Multiply the numerator and denominator by x1n (if we multiply both the numerator and denominator by the same number, we are multiplying by 1 and haven’t
changed the fraction.)
• Use this new function to determine as x → ±∞, what happens to f (x).
Example 4. Determine the end behavior of the following functions:
(a) f (x) =
x2 −7x5 +3
2x5 +4x3 −x2
(b) f (x) =
8−5x2 −7x3
4x10 −5x11 +x
(c) f (x) =
5x30 −x12 +7
25−x25
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Definition. If as x → ∞, f (x) → a or as x → −∞, f (x) → a, then the line y = a is a
horizontal asymptote.
Note.
• A rational function has at most one horizontal asymptote. Functions that
are not rational may have two different horizontal asymptotes, one as x → ∞ and
one as x → −∞.
• A function CAN cross it’s horizontal asymptote!
Horizontal Asymptotes: Let r(x) =
p(x)
q(x)
be a rational function of x, where
p(x) = pm xm + pm−1 xm−1 + · · · + p0 ,
q(x) = qn xn + qn−1 xn−1 + · · · + q0
with m the degree of p(x) and n the degree of q(x). Then we have three cases:
1. If m < n (i.e., the degree of the denominator is larger than the numerator), then
the horizontal asymptote is y = 0.
2. If m = n (i.e., the degree of the denominator is the same as the numerator), then
.
the horizontal asymptote is y = pqm
n
3. If m > n (i.e., the degree of the denominator is smaller than the numerator), then
there is no horizontal asymptote.
Example 5. Find the horizontal asymptotes of the following functions:
(a) f (x) =
x3 −x11 +x2
5x5 −3x2 −7
(b) f (x) =
33x95 −5x73 +10
15−10x43 −11x111 .
(c) f (x) =
3x5 −4x2 +4
x3 −5x5 +7x+15
(d) f (x) =
1
xn ,
where n is a positive integer.
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Vertical Asymptotes
Example 6. Graph the function f (x) =
1
x+2 ,
and find its vertical asymptote.
Definition. We say that a function y = f (x) has a vertical asymptote at x = a if
the numbers |f (x)| become arbitrarily large as x approaches the value a from the right
or the left. In other words, as x → a, f (x) → ±∞.
Finding Vertical Asymptotes of r(x) =
p(x)
q(x) :
1. First cancel any common factors out of
p(x)
q(x) .
2. Once p(x)
q(x) is in lowest terms, this rational function will have a vertical asymptote
at all values of x for which q(x) = 0. Therefore, set q(x) = 0, and solve for x.
Note.
• A rational function can have many different vertical asymptotes, but it
only has one horizontal asymptote. (Functions that are not rational, may have
two different horizontal asymptotes, one as x → ∞ and one as x → −∞).
• A function CANNOT cross it’s vertical asymptote (it’s undefined at this x-value).
Example 7. Find the vertical asymptotes for the following functions
1. f (x) =
3x3 +17x2 −28x
2x3 −19x2 +9x
2. f (x) =
x−1
x2 −x
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Holes in the Graph
What happens to the factors we can cancel out from the numerator and denominator?
For example, what does the graph f (x) = xx−1
2 −x look like?
Finding Holes of r(x) =
p(x)
q(x) :
1. First cancel any common factors out of
p(x)
q(x) .
2. The rational function will have a hole at a if the original denominator was zero
when x = a, but the simplified denominator is no longer zero. Plug x = a into
the simplified fraction to find the y-value of the hole.
Note. For rational functions r(x) = p(x)
q(x) , if q(a) = 0 then r(x) has either a hole or a
vertical asymptote when x = a. Simplify the function r(x) and
1. If the denominator is still zero when x = a, then it’s a vertical asymptote.
2. If the denominator is no longer zero when x = a, then it’s a hole.
Example 8. Does the function f (x) =
5(x−1)(x−3)2 (x+4)
3(x−1)(x−3)(x+4)2
have any holes? If so, where?
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Examples
Example 9. For the following rational functions, find the domain, x-intercepts, yintercepts, holes, vertical asymptotes, horizontal asymptotes, and end behavior.
1. f (x) =
5(x−1)(x−3)2 (x+4)
3(x−1)(x−3)(x+4)2
Domain:
x-intercept(s):
y-intercept:
Holes:
Vertical Asymptotes:
Horizontal Asymptotes:
as x → ∞, f (x) →
2. f (x) =
,
as x → −∞, f (x) →
,
as x → −∞, f (x) →
3x(2x−5)(x−3)
(x−7)(3x−4)(x−3)
Domain:
x-intercept(s):
y-intercept:
Holes:
Vertical Asymptotes:
Horizontal Asymptotes:
as x → ∞, f (x) →