Rational Functions

f(x) = p(x)/q(x), q(x) cannot equal 0

f(x) = 1/x this is a simple rational, it is also the reciprocal of a linear functions.

therefore...

f(x) = 1/x^2 is reciprocal of a quadratic f(x) = 1/x^3 is a reciprocal of a cubic and etc.

Special Skills Required

Things you would need to know when dealing with rationals are...

- Finding a Vertical and/or Horizontal Asymptote

- Find any y-intercepts, zero's, holes (if there are any)

- State the Domain and Range

- Graph a rational function

- Factor into factored form

Key Features

- As stated above, the denominator cannot equal 0, anything being divided by 0 causes problems.

Horizontal Asymptote (H.A.)

- if n < m, H.A. is y =0

- if n = m, H.A. is y = coefficient of x^n/ coefficient of x^m

- if n > m, there is no H.A.

Vertical Asymptote (V.A.)

- find the zero's of the denominator

Y-intercept

- Lets x=0

X-intercept

-Let y=0

Example

f(x) = x^2 - 3x - 10 / x^2 + 9x + 14

Factoring

f(x) = x^2 - 3x - 10/x^2 + 9x + 14

Quadratic formula on top and bottom

f(x) = (x-5) (x+2) /(x+7)(x+2)

the (x+2) cancels each other out

f(x) = (x-5)/(x+7) , x cannot equal 7

Horizontal Asymptotes

- n = m so...

y= coefficient of x^n/coefficient of x^m

y= 1/1

y= 1

Vertical Asymptotes

- solve for the zero's of the denominator

x= -7

Y-intercept

-let x = 0

y= (0-5)/(0+7)

y= -5/7

X-intercept

-let y = 0

0= (x-5)/(x+7)

0= x-5

x=5

Holes

x= -2

Graph

Domain and Range

D= {x| x cannot equal -7, xE R}

R= {y| y cannot equal 1,> yER}

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