Rational Functionswith Homero Simpson

1. Define rational functions. 2. Find the domain of a rational

function. 3. Find the asymptotes of a rational

function. 4. Draw the graph of a rational function.

Objectives

AsymptoteOne way to study the behavior of a function when the values tend to infinity or at the points where the function is not defined (isolated points) is to compare the function with a straight line, so we say that a line is an asymptote of function when the graph of the function and the line remain very close. Depending on the line as we have three types of asymptotes: Vertical, Horizontal and Oblique.

But what is the definition of a rational function? It is the function of the form

R x p xq x

( ) ( )( )

Where p (x) and q (x) are polynomial functions q (x) is not zero.

The domain consists of all real numbers except those for which the denominator q (x) is 0. Polynomial is the sum of several

monomials.

Monomial: algebraic expression in which letters, numbers and symbols are used

Domain: The set of values for which a function is defined

Codomain: One function is the set involved in that function.

YXf : Y

But it's function? It is the term used to

indicate the relationship or correspondence between

two or more quantities.

Example: Find the domain of the following rational functions:

Real Numbers: include rational numbers (such as 31, 37) and to those Irrational numbers can not be expressed fractionally and have infinite decimal places.

DefiniciónIf x tends to (x ) ó x -, and the value of R (x) to a fixed number L is about, then the line y = L is a horizontal asymptote of the graph of R.

y = L

y = R(x)

x

horizontal asymptote

Asymptote: A function whose graph representation is in the form of a straight

line or parabola and its trajectory is approaching a curve.

Horizontal Asymptote: It's called horizontal asymptote. The value (Real number) tends to F (x) to increase (or decrease) the x indefinitely.

y = L

y = R(x)y

x

x

y = L

y = R(x)

y

x

If x approaches a real number c, and the value of |R(x)| , “spproaches infinity ", then the line x = c is a vertical asymptote of the graph of R.

y

x

Vertical asymptote x = c

xVertical asymptotes: vertical lines are to which the function is approaching indefinitely without ever cutting.

Infinity: Any reference to an amount no limit or end, as opposed to the concept of finitude.

Finito: A group with a finite number of elements.

definition

If an asymptote is neither horizontal nor vertical is called oblique asymptote.

y

x

Oblique asymptote

Oblique asymptotes are straight equation

nmxY xxfm )(lim

x

For values of x increasing (in absolute value), the points on the line and the graph of the function are increasingly coming.

Theorem of Vertical Asymptotes

Asymptote: It tells a function f (x) to a straight t whose distance from the curve tends to zero when x tends to infinity or x tends to a point a.

A rational function in reduced form, has a vertical asymptote at x = r, si x –r is a factor of the denominator q(x); is, q(r )= 0 .

The line x=a is vertical asymptote (AV) de f(x) if limx->a+ f(x) = inf olimx->a- f(x) = inf.

EYE: To x = r is a vertical asymptote q(r) = 0 but p(r) ≠ 0.

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

2

3(a) ( )1

R xx

3( 1)( 1)x x

The graph has vertical asymptotes at : x = - 1 y en x = 1

2

3(b) ( )12

xR xx x

3( 3)( 4)

xx x

1

4x

The graph has a vertical asymptote at x = - 4

2

5(b) ( )1

xR xx

2 1 0x x i R

The graph has no vertical asymptotes

2

4(c) ( )12

xR xx x

4

( 3)( 4)x

x x

13x

The graph has a vertical asymptote has x = 3

Theorem horizontal and oblique asymptotes - Consider the rational function

R x p xq x

a x a x a x ab x b x b x bn

nn

n

mm

mm( ) ( )

( )

11

1 0

11

1 0

1. If n < m, then the line y = 0 is an horizontal asymptote of the graph of R.

2. If n = m, then the line y = an / bm is an horizontal asymptote of the graph of R.

wherein the degree of the numerator is the degree of the denominator n is m.

http://www.coolmath.com/graphit/

The best way to have a reference of how to graph is

using

It is very easy to use

3. Ifi n = m + 1, then the line y = ax + b is an oblique asymptote of the graph of R, where ax + b is the quotient of the division between p (x) y q (x).

4. If n > m + 1, he graph of R is not linear or horizontal or oblique asymptotes.

Horizontal asymptotes: We tend to indicate the function when x is large or very small mus also are parallel to the axis OX lines. Written y= asymptotic value. Oblique Asymptotes: A rational function has oblique asymptote when the degree of the numerator is greater than the degree of the denominator unit.

2

3 2

3 4 15(a) ( )4 7 1x xR x

x x x

The horizontal asymptote is: y = 0

2

2

2 4 1(b) ( )3 5x xR xx x

The horizontal asymptote is; y = 2/3

Example Find the horizontal or oblique asymptote of the graph of the function, if any.

The oblique asymptote is; y = x + 62 4 1(c) ( )

2x xR xx

2

2

6 2 4 1

- 2

6 1

- 6 12

13

xx x x

x x

xx

QUESTIONS?

THANK YOU