Rational Functions,

1
Chapter 5 Functions
5.6
Rational Functions, Equations, and Inequalities
5.6.1
MATHPOWERTM 11, WESTERN EDITION
2
Rational Expressions
Any algebraic expression that can be written as
the quotient of two polynomials is called a
rational expression.
Since division by zero is undefined, the
denominator cannot equal zero. Thus,
restrictions must be placed on the value of the
variable so that the denominator will not be
equal to zero. These restrictions are called the
Non-Permissible Values (NPVs).

• To solve a rational equation
• Identify the values of the variable for which the
  
• rational expressions in the equation are not
  defined.
• Multiply each side by the common denominator
• and solve the equation that results.
• Reject any roots that are values of the variable
  for
• which the rational expressions are not defined.

5.6.2
3
Solving Rational Equations
1. Find the LCD. 2. Multiply every term by the
LCD. 3. Simplify the equation.
LCD 12
4
2
3
12
12
12
12
4(2)
2(2x 3)
3x
12(1)
3x 12 8 4x 6 3x - 4x 14 - 12
-x 2 x -2
5.6.3
4
Solving Rational Equations
1. Factor first.
2. Find the LCD.
(x - 3)(x - 4)
(x - 3)(x - 4)
3. Multiply every term by the LCD.
x ? 3, 4
3
5(x - 4)
4(x - 3)
3 5x - 20 4x - 12 5x - 17 4x - 12
x 5
5.6.4
5
Solving Rational Equations
Solve
1. List NPVs
x ? 0, -2
2. Multiply both sides of the
equation by the LCD,
4(x 2)
3x
5x2 10x
4x 8 3x 5x2 10x 0
5x2 3x - 8 0 (5x 8)(x
- 1) 5x 8 0 or x - 1 0
x -1.6 or x 1
x(x 2).
Therefore, the solution is x -1.6 or x 1.
5.6.5
6
Solving Rational Equations
x ? -2, -3
- (2x2 1)
3(x 2)
3x(x 3)
3x2 9x - 2x2 - 1 3x 6 x2
6x - 7 0 (x 7)(x - 1) 0 x 7
0 or x - 1 0 x -7 or
x 1
Therefore, the solution is x -7 or x 1.
5.6.6
7
Rational Functions
A rational function is a function of the form
where g(x) and h(x) are polynomials and h(x) ? 0.
Graph
Note The function is undefined when x 2. On a
graph, this shows up as an asymptote.
An asymptote is a line that a curve approaches
but never touches. The line x 2 is the
vertical asymptote. The line y 0 is the
horizontal asymptote.
Asymptotes
The domain is the set of real numbers, x ?
2. The range is the set of real numbers, y ? 0.
5.6.7
8
Graphing Rational Functions
Graph
Vertical asymptotes are at x -2 and x 2.
The line y 0 is the horizontal asymptote.
The domain is the set of real numbers, but x ?
-2 and x ? 2.
The range is y gt 0 and
5.6.8
9
Graphing Rational Equations
Solve
x ? -2, -3
The solution is the point of intersection of the
two graphs.
The solution is x -7 or x 1.
5.6.9
10
Graphing Rational Expressions
Graph
Note At the point of x -2, there is an open
circle, making the graph discontinuous.
The domain is the set of real numbers but x ? -2.
The range is the set of real numbers but y ? -4.
5.6.10
11
Graphing Rational Functions
Graph
The break in the graph is at x 3.
The domain is the set of real numbers but x ? 3.
The range is the set of real numbers but y ? -1.
5.6.11
12
Solving Rational Inequalities Graphically
Solve
Find the values of x for which the graph is on
or above the x-axis.
(1, 0)
(4, 0)
x 4
-2 lt x lt 1
The graph is above the x-axis when -2 lt x and x lt
1 or x gt 4.
5.6.12
13
Solving Rational Inequalities
Graphically
Graph
x gt -1
Therefore, the solution is x gt -1.
5.6.13
14
Solving Rational Inequalities Graphically
Solve
Find the value of x for which the graph is
below the line y x 3.
The graphs intersect at x -8, x -2, and x 9.
The graphs approach the asymptotes of x -6
and x 8.
-8 lt x lt -6
-2 lt x lt 8
x gt 9
Therefore, the solution is -8 lt x and x lt -6
y x 3
-2 lt x and x lt 8
x gt 9.
5.6.14
15
Solving Rational Inequalities
Graphically
Solve
x lt -2
-1 lt x lt 0
Therefore, the solution is -2 gt x or -1 lt x lt 0.
5.6.15
16
Comparing a Function and its Reciprocal
Compare y x2 - 1 and
y x2 - 1
The graph of y x2 - 1 has zeros of x -1 and
x 1 which are the asymptotes of the second
graph.
The domain is the set of real numbers.
The domain of the second graph is the set of
real numbers but x ? -1 and x ? 1.
The range is y gt -1.
The range is y gt 0 and y lt -1.
5.6.16
17
Assignment
Suggested Questions
1-12, 21, 25, 27, 31, 35, 44, 48, 85, 95,
97, 103, 106, 114, 125
Page 308-312 51, 53, 60, 61, 63, 69, 74
5.6.17