I will assume that since the variable is called #x#, we are restricting ourselves to #x in RR#. If so, #RR# is the domain, since #f(x)# is well defined for all #x in RR#.

The highest order term is that in #x^4#, ensuring that:

#f(x) -> +oo# as #x -> -oo#

and

#f(x)->+oo# as #x -> +oo#

The minimum value of #f(x)# will occur at one of the zeros of the derivative:

#d/(dx) f(x) = 4x^3-12x^2+8x#

#= 4x(x^2-3x+2)#

#= 4x(x-1)(x-2)#

...that is when #x = 0#, #x = 1# or #x = 2#.

Substituting these values of #x# into the formula for #f(x)#, we find:

#f(0) = 1#, #f(1) = 2# and #f(2) = 1#.

The quartic #f(x)# is a sort of "W" shape with minimum value #1#.

So the range is #{ y in RR: y >= 1 }#