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Rational Functions and Domains
• A rational expression is given by
P  x
Q  x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
• Example 1
The following are examples of rational expressions:
x3
x  3x  4
2
x4
5x
4
6
x 5
Note that 8x is a rational expression
since it can be written in the form
8x
8x
1
• A rational function is given by
f ( x) 
P  x
Q  x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
• Example 2
The following are examples of rational functions:
f ( x) 
g ( x) 
x5
x4
x4
x  5x  3
2
• 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
Determine the domain of the function.
x 3
f ( x)  2
x 4
2
Set the denominator
x 4 0
equal to zero …
… and solve.
 x  2 x  2  0
x  2, 2
The solutions are
x  2
Dom f : All real numbers,
x  2
In interval notation
 , 2   2,2   2, 
• Example 4:
Determine the domain of the function.
x  7x  2
f ( x) 
2
x  3x
2
Set the denominator
x  3x  0
equal to zero …
x  x  3  0
2
… and solve.
x  3,0
The solutions are
x  3,0
Dom f : All real numbers,
x  3,0
In interval notation
 , 3   3,0   0, 
• Example 5:
Determine the domain of the function.
2x
f ( x)  2
x 9
Set the denominator
equal to zero.
x 9  0
2
There is no real number for x that will make
this equation true.
Dom f : All real numbers
In interval notation
 ,  