Section 3.7
Proper Rational Functions
Objectives:
1. To identify proper rational
functions in reduced form.
2. To identify horizontal and vertical
asymptotes, domains, and
ranges of reduced proper rational
functions.
3. To graph reduced proper rational
functions.
Definition
Rational function A function
P(x )
f(x) such that f ( x ) =
Q (x )
where P(x) and Q(x) are
polynomials and Q(x) ≠ 0.
Examples of rational functions are
x2 – 5
f(x) = x + 1 and
3x + 2
g(x) = x3 – 6x2 + 11x - 6 .
As with reciprocal functions, you can
find the domain by excluding values
where the denominator is zero. To
evaluate a rational function,
substitute the given domain value and
simplify to find the range value.
EXAMPLE 1 Evaluate f(x) and g(x)
for x = 0 and x = 1/2. Give the domains.
x2 – 5
f(x) =
x+1
02 – 5 -5
f(0) =
=
= -5
0+1
1
(1/2)2 – 5 -19/4
f(1/2) = 1
= 3 = -19/6
/2 + 1
/2
EXAMPLE 1 Evaluate f(x) and g(x)
for x = 0 and x = 1/2. Give the domains.
3x + 2
g(x) = 3
x – 6x2 + 11x - 6
3(0) + 2
2/ = -1/
g(0) = 3
=
-6
3
0 – 6(0)2 + 11(0) - 6
7/
1
3(
/
)
+
2
2
2
1
g( /2) = 1 3
= 15
1
2
1
( /2) – 6( /2) + 11( /2) - 6 - /8
= -28/15
EXAMPLE 1 Evaluate f(x) and g(x)
for x = 0 and x = 1/2. Give the domains.
Let x + 1 = 0 to determine when the
denominator of f(x) will be 0.
x+1=0
x = -1
D = {x|x  -1}
EXAMPLE 1 Evaluate f(x) and g(x)
for x = 0 and x = 1/2. Give the domains.
Let x3 – 6x2 + 11x – 6 = 0 to determine
when the denominator of g(x) will be 0.
Possible rational zeros are ±1, ±2, ±3, ±6.
Use synthetic division to factor the
polynomial.
x3 – 6x2 + 11x – 6 = 0
(x – 1)(x – 2)(x – 3) = 0
D = {x|x  1, 2, 3}
Definition
Proper rational function A
rational function in which the
degree of the numerator is less
than the degree of the
denominator.
A reduced rational function is
a function in which the
numerator and denominator
have no common factors.
1
EXAMPLE 2 Graph f(x) =
.
x+5
x+5=0
x = -5
D = {x|x ≠ -5}
1
1
f(0) =
= , plot (0, 1/5)
0+5 5
There are no x-intercepts.
1
EXAMPLE 2 Graph f(x) =
.
x+5
1
1
f(-6) =
= = -1
-6 + 5 -1
1
1
f(-4) =
= =1
-4 + 5 1
1
EXAMPLE 2 Graph f(x) =
.
x+5
Vertical asymptotes occur at
any x-value that makes the
denominator of a reduced
rational function equal to zero.
Horizontal asymptotes is y = 0
for every proper rational
function. Horizontal
asymptotes tell what happens
as x approaches ±.
x+2
EXAMPLE 3 Graph g(x) = x2 – 3x + 2
x2 – 3x + 2 = 0
(x – 2)(x – 1) = 0
x = 2 or x = 1
g(0) = 2/2 = 1; plot (0, 1)
x + 2 = 0 at x = -2; (-2, 0)
x+2
EXAMPLE 3 Graph g(x) = x2 – 3x + 2
g(3) = 5/2 = 21/2; plot (3, 21/2)
g(3/2) = -14; plot (3/2, -14)
x+2
EXAMPLE 3 Graph g(x) = x2 – 3x + 2
-2
2
-5
-10
-15
4
Homework:
pp. 155-157
►A. Exercises
For each graph below, identify the
intercepts, asymptotes, domain, and
range.
►A. Exercises
1.
►A. Exercises
3.
►A. Exercises
For each function give the y-intercept, the
domain, and any vertical asymptotes.
6
5. f(x) =
x+2
►A. Exercises
For each function give the y-intercept, the
domain, and any vertical asymptotes.
2 + 3x - 4
x
7. h(x) = 2
x + 4x – 12
►B. Exercises
Decide if each function above is proper
and reduced. If not, explain why. If it is,
give the x-intercept and the horizontal
asymptotes. Graph each function and
identify any graphs that are continuous,
odd, or even.
6
11. f(x) =
x+2
►B. Exercises
Decide if each function above is proper
and reduced. If not, explain why. If it is,
give the x-intercept and the horizontal
asymptotes. Graph each function and
identify any graphs that are continuous,
odd, or even.
x2 + 3x – 4
13. h(x) = 2
x + 4x – 12
■ Cumulative Review
Consider the following functions.
f(x) = x5 – x3
k(x) = tan x
g(x) = 3x2 + 1
p(x) = x6 – 4x
h(x) = [x]
q(x) = 1/(x2 – 4)
j(x) = cos x
r(x) = |x|
30. Which are odd functions?
■ Cumulative Review
Consider the following functions.
f(x) = x5 – x3
k(x) = tan x
g(x) = 3x2 + 1
p(x) = x6 – 4x
h(x) = [x]
q(x) = 1/(x2 – 4)
j(x) = cos x
r(x) = |x|
31. Which are even functions?
■ Cumulative Review
Consider the following functions.
f(x) = x5 – x3
k(x) = tan x
g(x) = 3x2 + 1
p(x) = x6 – 4x
h(x) = [x]
q(x) = 1/(x2 – 4)
j(x) = cos x
r(x) = |x|
32. Which are neither?
■ Cumulative Review
Let f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.
33. List all possible integer zeros.
■ Cumulative Review
Let f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.
34. Factor f(x).