Rational Functions

A rational function is a function of

the form

where g (x) 0

)(

)()(

xg

xhxf

Domain of a Rational Function

The domain of a rational function

is the set of real numbers x so that g (x) 0

Example

If

the domain is all real numbers except x = 0

xy

5

)(

)(

xg

xh

Domain of a Rational Function

If then the domain is the set of all real

numbers except x = -2If

then the domain is the set of all real numbers x > 5

2

6)(

2

x

xxxf

5

2)(

x

xxf

Domain of a Rational Function

23

6)(

2

x

xxf

then the domain is the set of all real numbers, because the denominator of this function is never equal to zero

If

Unbounbed Functions

A function f(x) is said to be unbounded in the positive direction if as x gets closer to zero, the values of f (x) gets larger and larger.

We write this as f (x) as x 0

(read f (x) approaches infinity as x goes to 0)

An Unbounbed Function

x

-1/10 500

-1/100 5,0000

-1/1000 5,000,000

1/1000 5000,000

1/100 5,0000

1/10 500

2

5

x

Note that as x approaches 0, f (x) becomes very large

The function

below is unbounded

2

5)(x

xf

Unbounbed Functions

Note that as x gets closer to zero, the values of f (x) gets smaller and smaller.

We say f (x) is unbounded in the negative direction. We write this as f (x) as x 0

(read f (x) approaches negative infinity as x goes to 0)

x-1/10 100

-1/100 10,000

-1/1000 1,000,000

1/1000 1,000,000

1/100 10,000

1/10 100

2

1

x

Unbounbed Functions

If as x gets closer to zero from the

left, the values of f (x) gets smaller

and smaller, and as x gets closer to

zero from the right, the values of f (x)

gets larger and larger, we say f (x) is unbounded in both direction

Unbounbed Functions

and write this as f (x) as x 0

(read f (x) approaches positive or negative infinity as x goes to 0)

Unbounbed Functions

xxf

1)(

Vertical Asymptote

For any rational function in lowest terms

and

a real number c so that h (c) 0 and g (c)= 0

the line x = c is called a vertical asymptote

The function

has a vertical asymptote X = 4

)(

)()(

xg

xhxf

4

2)(

x

xxf

Vertical Asymptote

2

6)(

2

x

xxxf

23

6)(

2

x

xxf

For the function the vertical

asymptote is x = -2

The function

has no vertical asymptote because the

denominator is never zero `

Horizontal Asymptote

If the degree of the denominator of a rational

function f (x) = h (x) / g (x) is greater than or

equal to the degree of the numerator of the

rational function, then f (x) has a horizontal asymptote.

Horizontal Asymptote

If the degree of the denominator is greater than

the of the degree of the numerator, then the

horizontal asymptote is y = 0

Example

y = 0 is the horizontal asymptote

5

3)(

2 x

xxf

Horizontal AsymptoteIf the degree of the denominator is equal to the

of the degree of the numerator, then the

horizontal asymptote is y = a where a is a

non zero real number

Example

y = 2 is the horizontal asymptote

4

2)(

x

xxf

Horizontal Asymptote

4

2)(

x

xxf

Horizontal asymptote is y = 2

23

6)(

2

x

xxf

Horizontal asymptote is y = 0

Examples of horizontal asymptotes

Slant Asymptote

If the degree of the numerator is greater to the

degree of the denominator, we have a slant

asymptote

Example of a function that has a slant asymptote

5

3)(

2

3

x

xxf

Please review problems solved in class before doing

your homework