Rational Functions
Dr. Philippe B. Laval
Kennesaw State University
Abstract
This handout is a quick review of rational functions. It also contains
a set of problems to guide the student through understanding rational
functions. This is to be turned in on Monday March 25.
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Definition and Examples
P (x)
where
Q (x)
P and Q are polynomials. We assume P and Q have no factors in common.
Definition 1 A rational function is a function of the form r (x) =
We notice immediately that r is not defined for every real number. The
domain of r is the set of real numbers which are not zeros of Q.
The notation x → a+ means that x is approaching a and is larger than
a. In other words, x is approaching a from the right. Similarly, the notation
x → a− means that x is approaching a and is less than a. In other words, x is
approaching a from the left.
Definition 2 (asymptote) Consider the function y = f (x).
1. The line x = a is a vertical asymptote for f if y → ∞ or y → −∞ as
x → a+ or x → a− .
2. The line y = b is a horizontal asymptote for f if y → b as x → ∞ or
x → −∞.
A rational function has a vertical asymptote wherever its denominator is
zero but its numerator is not zero. In other words, the line x = a is a vertical
asymptote for some rational function, if and only if a is a zero of the denominator
but not of the numerator. The graphs below illustrate the relationship between
a function and its asymptote.
In the graph below, the line x = 2 is a vertical asymptote. Here, we have
that y → −∞ as x → 2− and y → ∞ as x → 2+
1
In the graph below, the line x = 2 is a vertical asymptote. Here, we have
that y → ∞ as x → 2− and y → −∞ as x → 2+
In the graph below, the line x = 2 is a vertical asymptote. Here, we have
that y → ∞ as x → 2− as x → 2+
2
In the graph below, the line x = 2 is a vertical asymptote. Here, we have
that y → −∞ as x → 2− as x → 2+
The four graphs above illustrate the four possibilities for the behavior of a
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function near its vertical asymptote. The graphs below shows the behavior of
a function with respect to its horizontal asymptote. The horizontal asymptote
represents the behavior of a function when x → ±∞. In the graph below, the
line y = 2 is a horizontal asymptote.
Can you think of other possibilities for the position of a function with respect
to its horizontal asymptote?
When a function y = f (x) has a vertical asymptote at x = a, it means that
it is not defined at x = a since a is a zero of the denominator. Therefore, we
cannot evaluate f (a). According to the definition of an asymptote, we know
that y → ±∞ as x → ±∞. The question is to know whether y is approaching
−∞ or ∞. In order to find the answer, we evaluate the function at values of x
closer and closer to a to see if we can detect a pattern. For example, consider
1
. We see that f is not defined at x = 1. To find
the function y = f (x) =
x−1
what y is approaching as x → 1+ and x → 1− , we plug into the function values
of x close to 1. Which values we try is not too important. What matters is that
we try values of x close to 1 and getting closer and closer. To see what happens
to y as x → 1+ , we might try 1.1, 1.01, 1.001, 1.0001. The table below helps us
with the answer.
x 1.1 1.01 1.001 1.0001
y 10 100 1000 10000
We can see that as x gets closer to 1 from the right, y is getting larger and
larger. This suggests that y → ∞ as x → 1+ .
To see what happens to y as x → 1− , we might try 0.9, 0.99, 0.999, 0.9999.
The table below helps us with the answer.
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x 0.9
0.99
0.999
0.9999
y −10 −100 −1000 −10000
We can see that as x gets closer to 1 from the left, y is getting larger and
larger in absolute value, but it is also negative. This suggests that y → −∞ as
x → 1− .
Using the concepts above, answer the questions below.
1.1
Group work
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Example 3 Consider f (x) = . Answer the questions below, explain your
x
answers.
1. What is the domain of f ?
2. With the help of a table of values, determine the value f approaches when x
approaches ∞. An equivalent way of asking this is: find what f approaches
when x → ∞.
3. Same question when x → −∞
4. Obviously, f is not defined at 0. We wish to study the behavior of f as
x approaches 0. x can either approach 0 from the right (x is larger than
0 and approaches 0), in this case we write x → 0+ . x can also approach
0 from the left (x is smaller than 0 and approaches 0), in this case we
write x → 0− . Using a table of values, determine what f approaches when
x → 0+ , when x → 0−
5. Sketch the graph of f and its asymptotes.
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Example 4 Consider g (x) =
. Answer the questions below, explain your
x−2
answers.
1. What is the domain of g?
2. With the help of a table of values, determine the value g approaches when x
approaches ∞. An equivalent way of asking this is: find what g approaches
when x → ∞.
3. Same question when x → −∞
4. Determine what g approaches when x → 2+ , when x → 2−
5. Sketch the graph of g and of its asymptotes.
6. Questions 1 - 5 can also be answered by considering g as a transformation
of f (see 2.4). Show how you would do this by first determining which
transformation would produce g from f .
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Conclusion 5 The important result to remember is that for positive real num1
1
bers,
= small number and
= big number. Another
big number
small number
way of saying this is:
• A fraction whose denominator approaches ±∞ and whose numerator approached some constant, will approach 0.
• A fraction whose denominator approaches 0, whose numerator does not
approach 0, will go to infinity. It will be ∞ if the fraction is positive. It
will be −∞ if the fraction is negative.
2
Graphing a Rational Function
To sketch the graph of a rational function, the following information has to be
determined.
• domain
• x and y-intercepts
• vertical asymptotes and behavior of the function near the vertical asymptotes
• horizontal asymptotes
• end behavior
We look at each of these in details through an example. We will use the
x−2
in the subsections which follow to illustrate the concepts
function f (x) = 2
x −1
outlined above.
2.1
Finding the Domain
The domain is the set of real numbers for which a function is defined. In the
case of a rational function, it is the set of all real numbers which are not zeros
of the denominator. To find the domain of a real function, we must then look
for the real zeros of its denominator. This is when the tools learned in the
previous sections will be useful. In our case, it is easy. The denominator of f is
x2 − 1 = (x − 1) (x + 1). Its zeros are ±1. Thus, the domain of f is the set of
all real numbers except ±1.
2.2
Finding the Intercepts
• x-intercepts. The x-intercepts are the values of x for which f (x) = 0. A
rational function is 0 if and only if its numerator is 0. Thus, finding the
x-intercepts of a rational function is equivalent to finding the real zeros of
its numerator. In our case, the x-intercept is x = 2.
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• y-intercept. The y-intercept is the point where the graph intersects with
the y-axis. Thus, to find it, we set x = 0. In other words, we compute
f (0).
0−2
02 − 1
= 2
f (0) =
The y-intercept is y = 2.
2.3
Finding the Vertical Asymptotes
A rational function will have a vertical asymptote at the zeros of its denominator
(as long as they are not also zeros of the numerator). We still need to study
the behavior of the function on either side of the asymptote. That is, does the
function go to ∞ or −∞? Therefore, f has two vertical asymptotes. They are
the lines x = −1 and x = 1. Now, we need to determine what f approaches as
x → −1− , x → −1+ , x → 1− and x → 1+ . We know it will be either ∞ or −∞
since we have a vertical asymptote there. To find out which one it is, we simply
need to know the sign of the fraction there. If the fraction is positive, it will
approach ∞. Otherwise, it will approach −∞. This is best done with a table
of value. For example, if x → −1− , then x is close to −1 and less than −1. A
possible value is −1.01. We evaluate f (−1.01) = −149. 75. Thus, as x → −1− ,
f (x) → −∞. If x → −1+ , a possible value for x is −.99. Since f (−.99) =
150. 25, we conclude that as x → −1+ , f (x) → ∞. Similarly, if x → 1− , a
possible value for x is ..99. Since f (.99) = 50. 754, we conclude that as x → 1− ,
f (x) → ∞. Finally, if x → 1+ , a possible value for x is 1.01. Since f (1.01) =
−49. 254, we conclude that as x → 1+ , f (x) → −∞.
2.4
Finding the Horizontal Asymptotes.
The horizontal asymptotes are found by studying the behavior of the rational
function as x → ±∞. We first derive the general result. Consider the rational
function
an xn + an−1 xn−1 + ... + a1 x + a0
r (x) =
bm xm + bm−1 xm−1 + ... + b1 x + b0
Recall that when x → ±∞, a polynomial behaves like its leading term. Therean xn
. Thus, we have:
fore, r (x) will behave like
bm xm
Proposition 6 Let r (x) =
the following:
an xn + an−1 xn−1 + ... + a1 x + a0
. Then, we have
bm xm + bm−1 xm−1 + ... + b1 x + b0
1. If m > n, then r has horizontal asymptote y = 0. In this case, the end
behavior is determined by the asymptote, no further study is needed.
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an
. In this case, the end
bm
behavior is determined by the asymptote, no further study is needed.
2. If m = n, then r has horizontal asymptote y =
3. If n > m, then r has no horizontal asymptote. Further study is needed for
the end behavior.
Since f falls in case 1, y = 0 is the horizontal asymptote.
2.5
End Behavior
Like for polynomials, we wish to know the behavior of the rational function
when x → ±∞. We saw earlier that when the function has a horizontal asymptote, then we know the answer. The end behavior is given by the horizontal
asymptote. This part only concerns the case when there is no horizontal asymptote, that is when the degree of the numerator is higher that the degree of the
denominator. To use the notation of the previous subsection, n > m. Let
P (x)
. Since degree of P > degree of Q, we can perform long division.
r (x) =
Q (x)
The division algorithm tells us that there exists polynomials D (x) and R (x)
such that P (x) = Q (x) D (x) + R (x). Therefore,
P (x)
Q (x)
Q (x) D (x) + R (x)
=
Q (x)
Q (x) D (x) R (x)
+
=
Q (x)
Q (x)
R (x)
= D (x) +
Q (x)
r (x) =
where D (x) is a polynomial, and the degree of R is strictly less than the degree
R (x)
of Q. Then, as x → ∞,
→ 0. Therefore, r (x) will behave like D (x) for
Q (x)
large values of x.
Definition 7 If D (x) as above is linear, say D (x) = ax + b, then ax + b is
called a slant or an oblique asymptote for r (x).
Example 8 Find the end behavior of r (x) =
If we perform long division, we find that
r (x) =
x3 − 1
x+1
x3 − 1
x+1
= x2 − x + 1 −
8
2
x+1
Therefore, r (x) will behave like x2 − x + 1 when x → ±∞. This can be verified
by graphing both functions. In the graph below, r (x) is in blue, x2 − x + 1 is in
red. We can see that for large (not so large in fact), the blue graph and the red
graph are very close, and getting closer as |x| increases.
Figure 1:
2.6
Problems to Turn in
Example 9 For each function below, do the following:
• find the domain
• find the intercepts
• find the asymptotes
• find the end behavior
• sketch the graph and the asymptotes
1. r1 (x) =
3x + 3
x−3
2. r2 (x) =
x2 − 2
x+1
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3. r3 (x) =
4. r4 (x) =
x2 − x − 6
x2 + 3x
6x4
−3
x2
Example 10 Same question with f (x) =
What did this teach you?
3x2 − 3x − 6
. What do you notice?
x−2
Example 11 Find a rational function which has vertical asymptotes x = −1
and x = 2, and horizontal asymptote y = 1.
Example 12 Do # 60, 61, 62 and 65 on pages 275, 276
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