rational functions and domains where p(x) and q(x) are polynomials, q(x) ≠ 0. a rational...
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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
• 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
• A rational function is given by
( )P x
Q xf x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
• Example 2
The following are examples of rational functions:
5
4( )
x
xf x
2
4( )
5 3
x
xg x
x
• 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
• 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
The solutions are 2x
Dom f : All real numbers, 2x
In interval notation
, 2 2,2 2,
• 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
The solutions are 3,0x
Dom f : All real numbers, 3,0x
In interval notation
, 3 3,0 0,
• 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
Dom f : All real numbers
In interval notation
,
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