pre calc chapter 2 section 6. rational functions functions with the independent variable on the...

Pre Calc

Chapter 2 Section 6

Rational Functions

• Functions with the independent variable on the bottom of a fraction

f(x) =N(x)D(x)

Where N(x) & D(x) are polynomials and D(x) is not a zero Polynomial

Domain of a Rational Function

The denominator can not = 0

Lets look at f(x) =

As x approaches 0

x -1 -.5 -.1 -.0001 -.0000001 0

f(x) -1 -2 -10 -10000 -10000000 - ∞

x 1 .5 .1 .0001 .0000001 0

f(x) 1 2 10 10000 10000000 ∞

From the Left

From the Right

Range of a Rational Function

Lets look at f(x) =

As x approaches ∞

x 1 10 100 1000 1000000 ∞

f(x) 1 1 .01 .001 .000001 0

From the Left

From the Right x -1 -10 -100 -1000 -1000000 - ∞

f(x) -1 -.1 -.01 -.001 -.000001 0

What does that look like

10 5 5 10

1.0

0.5

0.5

1.0

f(x) =

10 5 5 10

2

3

4

5

Vertical Asymptotes

f(x) =N(x)D(x)

Where D(x) = 0 is a vertical asymptotes

Horizontal Asymptotes

f(x) =N(x)D(x)

If the degree of N(x) = n and the degree of D(x) = m

If n<m the horizontal axis is the horizontal asymptote

If n=m the line designated by y = the ratio of the leading coefficient N(x) over the leading coefficient of D(x)

If n>m there is no horizontal asymptote