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

,