Rational Functions: Exam Review

So Exams are coming up and basically this post is going to cover some things you need to know for Rational Functions. Things such as key features and skills you need to have when dealing with rational functions.


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}