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Section 5.1.2
Rational Functions
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Lesson Objective:
Students will:
• Transform functions in the form
into shifts of
• Be able to simplify complicated rational expressions.
ax by
x c+
=-1
.yx
=
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Lesson Objective:
Students will:
• Review negative exponents.
• Be able to simplify complicated rational expressions.
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Definitions
Rational Function: the ratio of two functions of the form:
( )( ) , ( ) 0
( )g x
f x h xh x
= ¹
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Example of a Rational Function
2
32 3
( )2
xf x
x x
+=
- -
g(x)
h(x)
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More Examples
21
( )2 5
f xx x
=+ +
2
3 22( )
x xp x
x-
=
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Not Examples
1( )f x
x=
2( )
1
xp x
x=
+
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5-14:
a: Write as a mixed number.
31
17
2017
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5-14:
b: Illustrate the technique shown in (a) on the
number .
52
7
197
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5-16:
Use the technique in 5-14 to change
3( ) 1
1g x
x= +
-
2( )
1x
g xx
+=
-
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5-16:
a: Sketch g(x) using transformations.
3( ) 1
1g x
x= +
-
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5-16:
b: Use the technique to change2 5
( )3
xh x
x-
=-
1( ) 2
3h x
x= +
-
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Definitions
Negative Exponents:
1, 0n
na a
a- = ¹
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Multiplication of Exponents
0
0
1
1
n n
n n
a a
a a
a
-
- +
=
=
=
g
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Eliminate Negative Exponents
Multiply the rational expression
by 1 or .n
n
aa
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Example
1 2
( )x y
f xx y
- -+=
+
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5-19: Simplify each of the following expressions using concept of removing negative exponents.
a:1 2
2 1 2
x y yx y xy
-
- -
++
3 2 2
3 31xy x y
x y+
+
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5-19:
b:
2 6
3
11x y
xy+
+
2 3 3
3
x y yy x
-
-
++
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Assignment
Pg 235 #5-27 TO 5-28
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Assignment
Pg 235 #5-22 TO 5-26