Question 725321
given the rational function
r(x)=(x^3-x^2)/(x^3-3x-2)
find the x and y intercepts
find the vertical and horizontal asymptotes
sketch the graph of r(x)
state the domain and range of r(x)
***
y-intercept:
set x=0
y=0
.. 
x-intercept
set x=0
x^3-x^2=0
x^2(x-1)=0
x=0 (multiplicity 2)
or
x=1
..
Horizontal asymptotes:
When degree of numerator and denominator the same, divide lead coefficient of numerator by lead coefficient of denominator to determine horizontal asymptote.
Horizontal asymptote: y=1/1=1
..
Vertical asymptotes:
set denominator=0, then solve for x-values which make the function undefined.
x^3-3x-2=0
use rational roots theorem or graphing calculator to find x-values:
x≠-1 (multiplicity 2)
and
x≠2
Vertical asymptotes: x=-1 ,2
..
Domain: (-∞,∞)
Range: (-∞,∞)

Number Line:
<...+..-1...+....0...-....1..-....2...+....>
see graph below


{{{ graph( 300, 300, -10, 10, -10, 10,(x^3-x^2)/(x^3-3x-2)) }}}