Download Chapter 8: Rational Functions

8.1 Simplify Rational Functions (Page 1 of 25)
Chapter 8
Rational Functions
8.1 Simplifying Rational Expressions; Finding the
Domain and Range of Rational Functions
Rational Function
If P(x) and Q(x) are polynomials, then a rational
function f is a function that can be written in the form
f (x) =
P(x)
, where Q(x) π 0 .
Q(x)
Some examples of rational functions are
x 3 - 3x + 6
-2x 2 + 17
, g(x) =
,
f (x) =
x-8
5x - 1
h(x) = -
4
.
5x 3
Domain of a Rational Function
The domain of a rational function is the set of all real
numbers except where Q(x) = 0 .
Example 1 Find the domain of
x 3 - 3x
-2x 2 + 17
a. f (x) =
b. g(x) =
x-8
5x - 1
c. h(x) =
-4
5x 3
8.1 Simplify Rational Functions (Page 2 of 25)
Vertical Asymptote
A vertical asymptote is a vertical line that
the graph of a function gets arbitrarily close
to, but never touches nor crosses. For
example, the graph of f in figure 1 has a
vertical asymptote at x = 5. Vertical
asymptotes can only occur at the values of x
that make the denominator zero.
f (x) =
4
4
–4
x=5
Example 2
6x
x 2 - 25
1. Find the domain of g.
Let g(x) =
2. The sketch of g is shown in
figure 2. Write the equation of
each vertical asymptote on the
graph of g.
TI-83/84
ZStandard
3. What is the equation of the
horizontal asymptote?
4. What is the range of g?
x-5
8
Figure 1 Graph of f (x ) =
Figure 2 Graph of g (x ) =
3
6x
(x - 25)
2
3
x -5
8.1 Simplify Rational Functions (Page 3 of 25)
Example 3
The graph g(x) =
shown.
x+4
is
3x 2 - 7x - 10
1. Find the domain of g.
2. Find the equation of any vertical
asymptote in the graph of g.
3. Find the y-intercept in the graph
of g.
TI-83/84
ZStandard
8.1 Simplify Rational Functions (Page 4 of 25)
A Simplified Rational Expression
A rational expression is simplified if the numerator and
denominator have no common factors other than 1.
To Simplify a Rational Expression
1. Factor the numerator and denominator.
2. Divide out any common factors.
Example 4
x 2 - x - 12
Simplify the expression 3
x - 4 x 2 + 5x - 20
Example 5
1.
Simplify
a-b
b-a
2.
Simplify
x-5
5- x
3.
Simplify
2x + 7
7 + 2x
4.
Simplify
11x - 3
3 - 11x
8.1 Simplify Rational Functions (Page 5 of 25)
Example 6
Find the domain of each function and simplify the righthand side of the equation. Check your result with the
graphing and table functions of you calculator.
2x 2 - 6x - 20
1.
f (x) =
2x 2 - 50
2.
x 3 - 5x 2 + 6x
g(x) = 3
x - 2x 2 - 9x + 18
8.1 Simplify Rational Functions (Page 6 of 25)
Example 7
Let f (x) = 8x 3 - 125 and g(x) = 4x 2 - 25 .
Ê fˆ
1. Find the equation for Á ˜ (x) and simplify the rightË g¯
hand side of the equation.
Ê fˆ
2. Find Á ˜ (3) .
Ë g¯
8.2 Multiply and Divides Rational Expressions (Page 7 of 25)
8.2 Multiply and Divide Rational Expressions
The following properties assume division by zero does not
occur.
A C AC
Multiplying Rational Expressions ◊ =
B D BD
Dividing Rational Expressions
A C A D AD
∏ = ◊ =
B D B C BC
Example 1
Perform the indicated operation and simplify.
4 x 3 3x + 5
1.
◊
7x - 1 2x
2.
x2 - 9
4 x 2 - 25
◊ 2
2
2x - x - 10 x + 4 x - 21
8.2 Multiply and Divides Rational Expressions (Page 8 of 25)
3.
6x 2
4 x7
∏
x - 3 x +1
4.
Ê x 2 - 2x - 48 3x 2 - 9x ˆ 6x + 24
ÁË x 2 + 8x + 16 ∏ x 2 - 16 ˜¯ ◊ 5x + 30
8.2 Multiply and Divides Rational Expressions (Page 9 of 25)
Example 2
35x 2 - 25x
6- x
Let f (x) =
and
.
g(x)
=
2
4
x - 36
15x
1. Find an equation of the product function f ◊ g .
2. Find ( f ◊ g)(2) .
Example 3
81x 2 - 49
7 - 9x
Let f (x) = 2
and g(x) =
.
3x + 16x + 5
18x + 6
1.
Find an equation of the quotient function
2.
Find Á ˜ (3) .
Ë g¯
Ê fˆ
f
.
g
8.3 Add and Subtract Rational Expressions (Page 10 of 25)
8.3 Add and Subtract Rational Expressions
Let A, B and C be rational expressions and C π 0 . Then
A B A+ B
Adding Rational Expressions
+ =
C C
C
Subtracting Rational Expressions
A B A- B
- =
C C
C
Example 1
Perform the indicated operation and simplify.
x 2 + 5x
6
1.
+
x2 - 9 x2 - 9
2.
x2
2x - 8
+
x 2 + 7x + 10 x 2 + 7x + 10
8.3 Add and Subtract Rational Expressions (Page 11 of 25)
Example 2
Perform the indicated operation and simplify.
x
5
3.
x 2 - 25 x 2 - 25
4.
3x 2 + 5x
2x 2 + 7x + 15
x 2 + 10x + 21 x 2 + 10x + 21
Least Common Denominator (LCD)
The least common multiple (LCM) of two polynomials is
the simplest polynomial that has both polynomials as
factors. The least common denominator (LCD) of two
rational expressions is the LCM of the denominators.
To Find the LCD of Two Rational Expressions
1. Factor each denominator.
2. The LCD is the product of each distinct prime factor the
greatest number of time it occurs in a single
denominator.
8.3 Add and Subtract Rational Expressions (Page 12 of 25)
Example 3
Perform the indicated operation and simplify.
7
6
7 Ê ˆ 6 Ê ˆ
+
=
1.
˜¯ + ÁË ˜¯
3
3 Á
Ë
3x
2x 3x
2x
2.
3
5
+
x-5 x+3
3.
3
x
x 2 - 9 2x + 6
8.3 Add and Subtract Rational Expressions (Page 13 of 25)
Example 4
Perform the indicated operation and simplify.
3x - 1
5
1.
2x 2 - 7x - 4 x 2 - 8x + 16
2.
x
3
+
x-2 2- x
3.
6 ˆ
3
Ê x+2
+
ÁË 2
˜
x - x x 2 - 1¯ x 2 + x
8.3 Add and Subtract Rational Expressions (Page 14 of 25)
Example 5
x -1
x +1
Let f (x) =
and g(x) =
.
x +1
x -1
1. Find an equation of the difference function f - g .
2. Find ( f - g)(2) .
Example 6
3
x +1
and
.
g(x)
=
12x 3 - 22x 2 + 6x
30x 2 - 10x
Find an equation of the sum function f + g .
Find ( f + g)(2) .
Let f (x) =
1.
2.
8.5 Solve Rational Equations (Page 15 of 25)
8.4 Simplify Complex Rational Expressions
Complex Rational Expression
A complex rational expression is a rational expression
whose numerator and/or denominator is also a rational
expression. Some examples of complex rational
expressions are
x2 - 9
3
1
5
+ 2
2
x 2 + 2x + 1 ,
2x x ,
x-2
2x - 6
2 1
x
x +1
+ 2
4x + 4
5 x
x + 2 x - 4 x - 12
Simplify Complex Rational Expressions
– Method 1
1. Simplify the numerator and denominator of the
complex rational expression so that each only have one
rational expression.
2. Rewrite the division as multiplication
A
B = A ∏ C = A ◊ D = AD
C B D B C BC
D
3. Factor the numerator and denominator and simplify.
Example 1
12
Simplify x
8
x3
8.5 Solve Rational Equations (Page 16 of 25)
Example 2
x2 - 9
2
x
+ 2x + 1
Simplify
2x - 6
4x + 4
Example 3
3
1
+ 2
Simplify 2x x
2 1
5 x
8.5 Solve Rational Equations (Page 17 of 25)
Simplify Complex Rational Expressions
– Method 2
1. Create a simple rational expression by multiplying the
numerator and denominator of the primary rational
expression by the LCD of the “inside rational
expressions.”
2. Factor the numerator and denominator and divide out
any common factors.
Example 2
1.
12
Simplify x
8
x3
2.
x2 - 9
2
x
+ 2x + 1
Simplify
2x - 6
4x + 4
8.5 Solve Rational Equations (Page 18 of 25)
Example 3
Example 4
Let f (x) = 2 -
3
1
+ 2
Simplify 2x x
2 1
5 x
5
x
x +1
and g(x) =
.
+ 2
x+2
x + 2 x - 4 x - 12
1.
Find an equation of the quotient function
2.
Find Á ˜ (4) .
Ë g¯
Ê fˆ
f
.
g
8.5 Solve Rational Equations (Page 19 of 25)
8.5 Solve Rational Equations
Rational Equation
A rational equation is two equal rational expressions.
Steps to Solve a Rational Equation
1. Clear all denominators (i.e. make all denominators 1)
in the equation by multiplying both sides of the
equation by the LCD of all rational expressions.
2. Solve the simplified equation from step 1.
3. Check your solution(s). This step is particularly
important with rational equations because it is possible
to get what is called an extraneous solution – a false
solution that forces division by zero in the original
equation.
Example 1
3x - 1 2
7
Solve
.
+
=
4x
3x 6x
8.5 Solve Rational Equations (Page 20 of 25)
Example 2
Solve for x.
1 3
x
- =2 x
2
Example 3
Solve for x.
x-
1 3
=
2 x
Example 4
Solve for x.
2x
4
= 1+
x-2
x-2
8.5 Solve Rational Equations (Page 21 of 25)
Example 5
Solve for x.
-3x
4
= 2+
x+8
x+8
Example 6
Solve for x:
x
7
14
.
= 2
x + 2 5 - x x - 3x - 10
8.5 Solve Rational Equations (Page 22 of 25)
Example 5
x +1 x - 2
Let f (x) =
.
x-3 x+3
a. Without using the graphing capability of your
calculator, find the exact value of x when f (x) = 1 .
Then round those values to 2 decimal places.
b. Verify your results
graphically. That is, set
Y1 = f (x) and Y2 = 1 . Then
intersect the two graphs on
your calculator.
8.6 Modeling with Rational Functions (Page 23 of 25)
8.6 Modeling with Rational Functions
Example 1
The ski club plans to spend $1250 to charter a bus for a ski
trip. Each participant will pay an equal share of the bus
cost plus $350 for food, lodging and lift tickets.
1. Let C(n) represent the total cost (in dollars) for n
participants. Find the equation for C.
2. Let the function M (n) represents the mean cost per
student. Find the equation for M. What are the units of
M?
3. What is M(20)? Explain its meaning in this application.
4. What is the minimum number of participants required
so that each person’s cost is no more than $400?
8.6 Modeling with Rational Functions (Page 24 of 25)
Example 2
College textbook sales (in millions of
dollars) for various years are given in
the first table. The number (in
millions) of men and women enrolled
in U.S. colleges for various years is
given in the second table.
1. Let C(t) represent the total cost
of textbooks (in millions of
dollars) for t years since 1980.
Find the function C.
Year
1980
1985
1990
1995
2000
Year
1980
1985
1990
1995
2000
Textbook Sales
(millions of dollars)
1801
2410
3445
4310
4866
Enrollment (millions)
Women
Men
6.0
5.4
6.6
5.9
7.4
6.2
8.0
6.7
8.7
7.1
2. Let E(t) represent the total enrollment in U.S. colleges
for t years since 1980. Find the function E.
3. Let M (t) represent the mean amount of money spent
on textbooks per student. Find the function M.
4. Predict the mean amount of money that each student
will spend on textbooks in 2005.
5. When will the average student have to spend $500 on
textbooks.
8.6 Modeling with Rational Functions (Page 25 of 25)
Example 3
Suppose a student plans to drive from Miami, Florida to
Atlanta, Georgia. The speed limit is 70 mph in Florida and
65 mph in Georgia. The trip involves 473 miles in Florida
d
an 253 miles in Georgia. Recall that since r ◊t = d , t = .
r
1. If the student drives the speed limit, how much driving
time does it take?
2. Forget driving the speed limit - it takes too long. Find
the function T (a) that represents the time it takes the
student to complete the trip driving a mph above the
speed limit.
3. Find T (0) and explain its meaning.
4. Find T (8) and explain its meaning.
5. If the student wants the driving time to be 9 hours, how
much over the speed limits must she drive?