Rational Functions

Find the Domain of a functionDetermine the vertical asymptotes of a

rational functionDetermine the horizontal asymptotes of a

rational function

What is a rational function?

A rational function is a function of the form

R(x) =

Where p and q are polynomial functions and q is not a zero polynomial. The domain is the set of all real numbers except those for which the denominator q is 0

)(

)(

xq

xp

Finding the domain of a rational function

Find the domain of R(x) =

R(x) =

R(x) =

R(x) =

5

42 2

x

x

4

12 x

12

3

x

x

3

22 x

Lowest Terms

If R(x) = is a rational function and p and q

have no common factors, then the rational function R is said to be lowest terms.

When a rational function is in lowest terms the zeros if any of the numerator are the x intercepts of the graph of R and sill will play a major role in the graph of R

)(

)(

xq

xp

Properties of x1

x

1 Is the reciprocal Function from our library of functions in Chapter 2. The domain and the range is the set of all nonzero real numbers. The graph has no intercepts. The reciprocal functions is decreasing on the interval (- ∞, 0) and (0, ∞)

Graphing

The domain of H(x)= is the set of all real numbers except for 0. The graph has no y intercept because x can number be equal to 0. The graph has no x intercept because the equation H(x) = 0 has no solution. The graph is an even function so therefore is symmetric to the y axis.

2

1

x

Transformations

The same rules apply that we focused on in Chapter 2

y=

y= 1

)2(

1

)2(

1

2

2

x

x

Asymptotes

A horizontal asymptote when it occurs describes a certain behavior of the graph as x →∞ or as x →-∞, that is the end behavior. The graph of the function may intersect a horizontal asymptote.

A vertical asymptote, when it occurs describes a certain behavior of the graph when x is close to some number c. The graph of the function will never intersect a vertical asymptote.

Oblique Asymptote

If an asymptote is neither vertical nor horizontal it is called oblique. An oblique asymptote when it occurs describes the end behavior of the graph. The graph of a function may intersect and oblique asymptote.

Locating Vertical Asymptotes

A rational function R(x) = , in lowest terms will have a vertical asymptote x = r if r is a real number of the denominator q. That is, if x-r is a factor of the denominator q of a rational function. R(x) = in lowest terms, the R will Have the vertical asymptote x=r

)(

)(

xq

xp

)(

)(

xq

xp

Finding Vertical Asymptotes

Find the vertical asymptotes, if any of the graph of each rational function

R(x) =

F(x) =

G(x) =

H(x) = 214

9

1

1

3

4

2

2

2

2

2

xx

x

x

x

x

x

x

x

Horizontal Asymptotes

The procedure for finding horizontal asymptotes and oblique asymptotes is somewhat more involved.

To find these asymptotes we need to know how the values of a function behave as x→∞ or as x→ -∞.

If a rational function is proper, that is, if the degree of the numerator is less than the degree of the denominator, then as x→-∞ or as x→∞ the value of R(x) approaches 0. Therefore the line y=0 is a horizontal asymptote.

Rule

If a rational function is proper, the line y=0 is a horizontal asymptote of its graph

Finding Horizontal Asymptotes

Find the horizontal asymptotes if any of the graph of

14

12)(

2

xx

xxR

Improper Rational Function

If a rational function R(x) = is improper, that is the degree of the numerator is greater than or equal to the degree of the denominator.

We must use long division to write the rational function as the sum of a polynomial f(x) plus the proper rational function.

)(

)(

xq

xp

Possibilities

If f(x) = b a constant, then the line y = b is a horizontal asymptote of the graph of R

Example Find the horizontal of oblique asymptotes if any of the graph of

14

28)(

2

2

x

xxxR

Long Division First

14

28)(

2

2

x

xxxR

Possibility 2

If f(x) = ax+b, a≠ 0 then the line y = ax+b is an oblique asymptote of the graph of REx Find the horizontal of oblique

asymptotes if any of the graph of

1

3)(

23

24

xx

xxxR

Long Division

1

3)(

23

24

xx

xxxR

Possibility 3

In all other cases, the graph R approaches the graph f, and there are no horizontal of oblique asymptotes

Ex:

1

32)(

3

35

x

xxxG

1

32)(

3

35

x

xxxG