228
4.1
CHAPTER 4. RATIONAL FUNCTIONS
Graphs and Asymptotes
A rational function is a function that is the ratio of two polynomial func(x)
tions fg(x)
. The domain consists of all real numbers x such that g(x) 6= 0.
Example 4.1.1
Find the domain of the function f (x) =
notation.
x−2
.
x2 −x−6
Write your answer in interval
Solution.
The domain consists of all numbers x such that x2 − x − 6 6= 0. But this last
quadratic expression is 0 when x = −2 or x = 3. Thus, the domain is the set
(−∞, −2) ∪ (−2, 3) ∪ (3, ∞)
The Long-Run Behavior: Horizontal and Oblique Asymptotes
Given a rational function
am xm + am−1 xm−1 + · · · + a1 x + a0
.
f (x) =
bn xn + bn−1 xn−1 + · · · + b1 x + b0
We consider the following three cases:
Case 1: m = n
In this case, we can write
x n an +
f (x) = n ·
x bn +
an−1
x
bn−1
x
+ ··· +
+ ··· +
a1
xn−1
b1
xn−1
+
+
a0
xn
b0
xn
.
Now, for 1 ≤ k ≤ n we know that
1
→ 0 as x → ±∞.
xn
Thus, f (x) → abnn as x → ±∞. We call the line y = abnn a horizontal asymptote. Geometrically, the graph flatten out as x → ±∞.
Example 4.1.2
Find the horizontal asymptote of f (x) =
3x2 +2x−4
.
2x2 −x+1
Solution.
As x → ±∞, we have
3x2 + 2x − 4
2x2 − x + 1
x2 3 + x2 − x42
3
= 2·
1
1 →
x 2 − x + x2
2
f (x) =
4.1. GRAPHS AND ASYMPTOTES
Hence, y =
3
2
229
is a horizontal asymptote
Case 2: m < n
In this case, we can write
f (x) =
am−1
xm−1
bn + bn−1
x
am +
1
xn−m
+ ··· +
+ ··· +
a1
+ xam0
xm−1
b1
+ xb0n
xn−1
.
Hence, f (x) → 0 as x → ±∞. In this case, the x− axis is the horizontal
asymptote.
Example 4.1.3
Find the horizontal asymptote of f (x) =
2x+3
.
x3 −2x2 +4
Solution.
As x → ±∞, we have
2x + 3
− 2x2 + 4
2 + x3
1
= 2·
→0
x 1 − x2 + x43
f (x) =
x3
so the x−axis is the horizontal asymptote
Case 3: m > n
In this case, we can use long division of polynomial to write
f (x)
g(x)
r(x)
= q(x) + g(x)
where the degree of r(x) is less than that of g(x). As in case 2,
r(x)
g(x)
→ 0 as
f (x)
g(x)
x → ±∞ so that
− q(x) → 0 as x → ±∞. We call y = q(x) an oblique
asymptote. Geometrically, f (x) gets level out close to the oblique line as
x → ±∞.
Example 4.1.4
Find the oblique asymptote of f (x) =
2x2 −3x−1
.
x−2
Solution.
Using long division of polynomials, we find
f (x) = 2x + 1 +
1
.
x−2
Hence, f (x) − (2x + 1) → 0 as x → ±∞. Hence, y = 2x + 1 is the oblique
asymptote
230
CHAPTER 4. RATIONAL FUNCTIONS
Remark 4.1.1
It is possible for the graph to cross either the horizontal asymptote or the
oblique asymptote.
The Short-Run Behavior: Horizontal Intercepts/Vertical Asymptotes
We next study the local behavior of rational functions which includes the zeros and the vertical asymptotes.
The Zeros of a Rational Function
The zeros of a rational function are its x−intercepts. They are those
numbers that make the numerator zero and the denominator non-zero.
Example 4.1.5
Find the zeros of each of the following functions:
(a) f (x) =
x2 +x−2
x−3
(b) g(x) =
x2 +x−2
.
x−1
Solution.
(a) Factoring the numerator we find x2 + x − 2 = (x − 1)(x + 2). Thus, the
zeros of the numerator are 1 and −2. Since the denominator is different from
zero at these values, the zeros of f (x) are 1 and −2.
(b) The zeros of the numerator are 1 and −2. Since 1 is also a zero of the
denominator, g(x) has −2 as the only zero
Vertical Asymptotes
When the graph of a function either grows without bounds or decay without
bounds as x → a from either sides, then we say that x = a is a vertical
asymptote. For rational functions, the vertical asymptotes are the zeros
of the denominator. Thus, if x = a is a vertical asymptote then as x approaches a from either sides, the function either increases without bounds or
decreases without bounds. The graph of a function never crosses its vertical
asymptotes since the function is not defined there.
Example 4.1.6
Find the vertical asymptotes of the function f (x) =
2x−11
x2 +2x−8
Solution.
Factoring x2 + 2x − 8 = 0 we find (x − 2)(x + 4) = 0. Thus, the vertical
4.1. GRAPHS AND ASYMPTOTES
231
asymptotes are the lines x = 2 and x = −4
Graphing Rational Functions
To graph a rational function h(x) =
1.
of
2.
3.
4.
5.
f (x)
:
g(x)
Find the domain of h(x) and therefore sketch the vertical asymptotes
h(x).
Sketch the horizontal or the oblique asymptotes if they exist.
Find the x−intercepts of h(x) by solving the equation f (x) = 0.
Find the y−intercept, if it exists: h(0).
Draw the graph.
Example 4.1.7
Sketch the graph of the function f (x) =
x(4−x)
.
x2 −6x+5
Solution.
1. Domain = (−∞, 1) ∪ (1, 5) ∪ (5, ∞). The vertical asymptotes are x = 1
and x = 5.
2. As x → ±∞, f (x) → −1 so the line y = −1 is the horizontal asymptote.
3. The x−intercepts are at x = 0 and x = 4.
4. The y−intercept is y = 0.
5. The graph is given in Figure 4.1.1
Figure 4.1.1
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CHAPTER 4. RATIONAL FUNCTIONS
Case when numerator and denominator have common zeros
2 +x−2
We have seen in Example 4.1.5, that the function g(x) = x x−1
has a
common zero at x = 1. You might wonder what the graph looks like. For
x 6= 1, the function reduces to g(x) = x + 2. Thus, the graph of g(x) is a
straight line with a hole at x = 1 as shown in Figure 4.1.2.
Figure 4.1.2
4.1. GRAPHS AND ASYMPTOTES
233
Exercises
In Exercises 4.1.1 - 4.1.9 answer the following questions:
(a) Find the domain of existence.
(b) Find the horizontal/oblique asymptotes, if they exist.
(c) Find the vertical asymptote(s), if they exist.
(d) Find the intercepts.
(e) Graph.
Exercise 4.1.1
f (x) =
1
.
x2
Exercise 4.1.2
f (x) =
2
.
x+3
Exercise 4.1.3
f (x) =
−3
.
(x − 1)2
Exercise 4.1.4
f (x) =
x2
x
.
−1
Exercise 4.1.5
f (x) =
3x
.
x+1
f (x) =
4
.
x2 + 1
f (x) =
2x + 1
.
x+1
Exercise 4.1.6
Exercise 4.1.7
234
CHAPTER 4. RATIONAL FUNCTIONS
Exercise 4.1.8
f (x) =
2x2
.
3x2 + 1
f (x) =
x2 − x
.
x+1
Exercise 4.1.9
Exercise 4.1.10
Find the oblique asymptote of f (x) =
2x3 −1
.
x2 −1
Exercise 4.1.11
Write a rational function satisfying the following criteria:
Vertical asymptote: x = −1.
Horizontal asymptote: y = 2.
y−intercept: y = 3.
x−intercept: x = − 32 .
Exercise 4.1.12
Find the zeros of the rational function f (x) =
Exercise 4.1.13
Find the y−intercept of the function f (x) =
x2 +x−2
.
x+1
3
.
x−2
Exercise 4.1.14
Write a rational function with vertical asymptotes x = −2 and x = 1.
Exercise 4.1.15
Find the horizontal asymptote of f (x) =
2x−1
.
x2 +1
Exercise 4.1.16
Find the domain of the function f (x) =
x+4
.
x2 +x−6
Exercise 4.1.17
Find the horizontal asymptote of f (x) =
Exercise 4.1.18
Find the oblique asymptote of f (x) =
x2
.
3x2 −4x−1
x2 −1
.
2x
4.1. GRAPHS AND ASYMPTOTES
235
Exercise 4.1.19
Find the domain of (f ◦ g)(x) if f (x) =
Exercise 4.1.20
Find the domain of f (x) =
1
x+2
and g(x) =
4
.
x−1
2x−9
.
x3 +2x2 −8x
Exercise 4.1.21
Find the vertical asymptotes of f (x) =
x2 +1
.
x3 +2x2 −25x−50
Exercise 4.1.22
Find the horizontal asymptote of f (x) =
5x2 −x+2
.
2x2 +3x−7
Exercise 4.1.23
Find the horizontal asymptote of f (x) =
5x3 −x2 +2
.
2x4 +3x3 −7
Exercise 4.1.24
Find the oblique asymptote of f (x) =
Exercise 4.1.25
Find the x−intercepts of f (x) =
10x2 +7x+2
.
2x−3
x3 +2x2 −25x+50
.
x2 +x+1
Exercise 4.1.26
Sketch the graph of f (x) =
x2
.
x2 −1
Exercise 4.1.27
Sketch the graph of f (x) =
1−x2
.
x+1
Exercise 4.1.28
Sketch the graph of f (x) =
x2 −x+2
.
x−3
Exercise 4.1.29
The concentration C(in mg/dl), of a certain antibiotic in a patient’s bloodstream is given by
50t
C(t) = 2
t + 25
where t is the time (in hours) after taking the antibiotic.
(a) What is the concentration 4 hours after taking the antibiotic?
(b) In order for the antibiotic to be effective, 4 or more mg/dl must be present
in the bloodstream. When do you have to take the antibiotic again?
236
CHAPTER 4. RATIONAL FUNCTIONS
Exercise 4.1.30
A rare species of insect was discovered in the rain forest of Costa Rica. Environmentalists transplant the insect into a protected area. The population
of the insect t months after being transplanted is
P (t) =
45(1 + 0.6t)
.
(3 + 0.02t)
(a) What was the population when t = 0?
(b) What will the population be after 10 years?
(c) When will there be 549 insects?