math/calculus
posted by angie on .
find the domain and range of the function f(x)=x^29/4x^2+x

You have
(x3)(x+3) / x(4x+1)
The domain is all real numbers except where the denominator is zero. So, that would be all reals except 0 and 1/4
There is a horizontal asymptote at y = 1/4.
There are vertical asymptotes at x = 1/4 and x=0.
For x between 1/4 and 0, y > 0
As x gets big, y lies below the asymptote.
So, the range outside (1/4,0) is all reals less than 1/4.
y' = (x^2+72x+9)/(4x^2+1)^2
y' = 0 when x = .125
y(.125) = 143.75
So, the range is all reals
< 1/4 or >= 143.75