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Rational Functions and Domains
P x
Q x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
• A rational expression is given by
![Page 2: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/2.jpg)
• Example 1
The following are examples of rational expressions:
3
4
x
x
2
4
3 4
5
x x
x
Note that 8x is a rational expression since it can be written in the form
6
5x 8x
8
1
x
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• A rational function is given by
( )P x
Q xf x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
![Page 4: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/4.jpg)
• Example 2
The following are examples of rational functions:
5
4( )
x
xf x
2
4( )
5 3
x
xg x
x
![Page 5: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/5.jpg)
• The domain of a rational function is all real numbers except for those values for which the denominator is zero.
• To find the domain of a rational function
1. Set the denominator to zero and solve
2. The domain is all real numbers except for the solutions found in step 1
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• Example 3
Set the denominator equal to zero …
… and solve.
Determine the domain of the function.
2
3( )
4
xf x
x
2 4 0x
2 2 0x x
2,2x
![Page 7: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/7.jpg)
The solutions are 2x
Dom f : All real numbers, 2x
In interval notation
, 2 2,2 2,
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• Example 4:
Set the denominator equal to zero …
… and solve.
Determine the domain of the function.2
2
7 2( )
3
x xf x
x x
2 3 0x x 3 0x x
3,0x
![Page 9: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/9.jpg)
The solutions are 3,0x
Dom f : All real numbers, 3,0x
In interval notation
, 3 3,0 0,
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• Example 5:
Set the denominator equal to zero.
There is no real number for x that will make this equation true.
Determine the domain of the function.
2
2( )
9
xf x
x
2 9 0x
![Page 11: Rational Functions and Domains where P(x) and Q(x) are polynomials, Q(x) ≠ 0. A rational expression is given by](https://reader036.vdocuments.us/reader036/viewer/2022082821/5697c0141a28abf838ccd287/html5/thumbnails/11.jpg)
Dom f : All real numbers
In interval notation
,
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