How do you find domain and range for #f(x)=x/(x^3+8) #?

1 Answer
Sep 22, 2015

Domain = #RR-{-2}#
Range = #RR#


First factorize the denominator to find possible points of discontinuity.
Factorize it as a sum of 2 cubes, and then as a trinomial to get


The trinomial in the denominator is an irreducible quadratic and has no real roots.

Hence the only vertical asymptote is at #x=-2#

The horizontal asymptote would usually occur at #lim_(x-oo)f(x) =0#, since the denominator dominates the numerator and increases faster.
However in this case, 0 is in the range of the function since x = 0 in the numerator would set the output value y to 0.

So the domain is all real numbers x except -2, and the range is all real numbers y

Graphically :
graph{x/(x^3+8) [-10, 10, -5, 5]}