math problem

Question

State the domain and range of the function and sketch its graph:

 

s(x)= x^2-2x+1/x^3-3x^2

Answer 1

s(x) = (x-1)^2/(x^2(x-3))

x = any number except 0 or 3, as they make the denominator = 0 which is not allowed as it makes s(x) undefined

x=0 and x =3 are vertical asymptotes

range = all numbers

even though it seemingly is except s(x) = 0 which is a horizontal asymptote

0 is the ratio of the highest degreed term in the numerator over the highest degreed term in the denominator = x^2/ x^3 = 1/x which appraches zero as x approaches infinity or negative infinity

but

s(x) = 0= (x-1)^2/(x^2(x-3))

x^2 -2x +1 = 0

x= 1

even though to the right and left, y=0 is an asymptote

it may helpt to graph the function