Math 1014: Precalculus with Transcendentals
Ch. 3: Polynomials and Rational Functions
Sec. 3.5 Rational Functions and Their Graphs
I.
Rational Functions
A. Definition
Rational Functions are quotients of polynomial functions. This means that rational functions can be expr
f (x) =
p(x)
, where p and q are polynomial functions and q(x) ≠ 0 .
q(x)
B. Domain of Rational Functions
1. Definition:
The domain of a rational function is the set of all real numbers except the x-values that make the den
zero.
2. Examples
Find the domain of the rational functions
a.
f (x) =
4
x−5
b.
g(x) =
4x
x − 81
c.
h(x) =
3x − 6
x −x−2
d.
t(x) =
x+5
x 2 + 25
2
2
II.
Vertical Asymptotes and Horizontal Asymptotes of Rational Functions
A. Arrow Notation
Symbol
Meaning
+
x→a
x → a−
x→∞
x → −∞
x
x
x
x
approaches
a from the right
approaches a from the left
approaches infinity; i.e., x increases without bound
approaches negative infinity; i.e., x decreases without bound
B. Definitions
1. Vertical Asymptote
x = a is a vertical asymptote of the graph of a function f if f (x)
increases or decreases without bound as x approaches a .
The line
y
y
y
y
x
f
f
f
x
x
x=a
x=a
As x → a + ,
f (x) → ∞
Thus, as
x
f
x=a
As x → a − ,
f (x) → ∞
x=a
As x → a + ,
f (x) → −∞
As x → a − ,
f (x) → −∞
x approaches a from either the left or the right, f (x) → ∞ or f (x) → −∞ .
2. Horizontal Asymptote
y = b is a horizontal asymptote of the graph of a function f if f (x)
approaches b as x increases or decreases without bound.
y
y
f
y
y=b
y=b
f
The line
f
x
y=b
As x → ∞,
As x → ∞,
f (x) → b
f (x) → b
Similar graphs as x → −∞.
x
x
As x → ∞,
f (x) → b
C. Graphical Example
y
10
9
8
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
x
-2
-3
-4
-5
-6
-7
-8
-9
-10
1) as x → −5, f (x) → ________
6) as x → −4 + , f (x) → ________
2) as x → 0 + , f (x) → ________
7) as x → 3− , f (x) → _________
3) as x → 0 − , f (x) → ________
8) as x → 3+ , f (x) → _________
4) as x → 5, f (x) → _________
5) as x → −4 − , f (x) → ________
9) as x → −∞, f (x) → _________
10) as x → ∞, f (x) → _________
State the equation(s) of the vertical asymptote(s) : ____________________
State the equation(s) of the horizontal asymptote(s) : ____________________
D. Algebraic Approach to Finding Asymptotes
1. Locating Vertical Asymptotes
p(x)
is a rational function in which p(x) and q(x) have no common
q(x)
factors and a is a zero of q(x) ,the denominator, then x = a is a vertical
asymptote of the graph of f .
a. If
f (x) =
b. Method
1) Factor numerator and denominator.
2) Cancel out any factors possible.
3) Set denominator equal to 0 and solve.
c. Examples
Find the vertical asymptote(s).
1)
f (x) =
4x 2
(x − 2)(x + 6)
4x 3
2) g(x) = 2
x − 3x
3)
t(x) =
x+4
x 2 + 36
4)
h(x) =
x+4
x 2 − 3x
2. Locating Horizontal Asymptotes
a. Let
f be the rational function given by
f (x) =
an x n + an−1 x n−1 + ...+ a1 x + a0
, an ≠ 0, bm ≠ 0 .
bm x m + bm−1 x m−1 + ...+ b1 x + b0
The degree of the numerator is n. The degree of the denominator is
m.
1) If n < m , y = 0 is the horizontal asymptote of the graph of f .
an
is the horizontal asymptote of the graph of f .
bm
2) If
n = m , the line y =
3) If
n > m , the graph of f has no horizontal asymptote.
b. Examples
Find the horizontal asymptote(s).
1)
f (x) =
4x 2
(x − 2)(x + 6)
2)
g(x) =
4x 3
x 2 − 3x
3)
t(x) =
x+4
x 2 + 36
4)
h(x) =
x+4
x 2 − 3x