SOLUTION: 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 rang

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)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
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

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