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For a rational function, find the domain and graph the function, identifying all of the asymptotes.
Solve applied problems involving rational functions.
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4.5 Rational Functions
Rational Function
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A rational function is a function f that is a quotient of two polynomials, that is,
where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x) 0.
( )( ) ,
( )p x
f xq x
Example
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Consider .
Find the domain and graph f.
The graph of the function is the graph of y = 1/x translated to the left 4 units.
Example
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Determine the domain of each of the functions shown below.
a. b.
| 0 , or ,0 0, x x | 0 , or ,0 0, x x
Example
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Determine the domain of each of the functions shown below.
c. d.
| 2 1 , or
, 2 2,1 1,
x x and x | 3 , or ,3 3, x x
Example
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Determine the domain of each of the functions shown below.
e. f.
| 1 2 , or
, 1 1,2 2,
x x and x | 1 , or , 1 1, x x
Vertical Asymptotes
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Example
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Determine the vertical asymptotes of the function.
2
2 3( )
4
xf x
x
Example
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Determine the vertical asymptotes of the function.
2
3
4( )
x x
f xx x
Hole located at (0, 4)
Horizontal Asymptotes
Determining a Horizontal Asymptote
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Notice how the horizontal asymptote intersects the graph.
Examples of Graphs with Horizontal Asymptotes and without
Horizontal Asymptote
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Example: Find the horizontal asymptote: 4 2
4
6 3 1( ) .
9 3 2
x xf x
x x
Graphing Rational Functions
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1. Determine the domain of the function.
2. Find the real zeros of the denominator and sketch any vertical asymptotes.
3. Find the horizontal or oblique asymptote, if there is one, and sketch it.
4. Use the graph on a calculator to sketch in the function.
Example
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Graph
2( ) .
3
xh x
x
Example
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Graph
2
3( ) .
2 5 3
xf x
x x
More Examples
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Graph the following functions.
a)
b)
c)
3( )
2
xf x
x
2
2
8( )
9
xf x
x
2
( )1
xf x
x
More Practice
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