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