Much like a rational function in #x#, we must ensure that the denominator is not #0#.
#3 - cos2x ne 0#
#3 ne cos2x#
Note that the range of the cosine function is #[-1, 1]# so this inequality always holds true. In other words, the denominator is never equal to #0# so the domain is #(-oo, oo)#.
#F# is a continuous function so it suffices to find the maximum and minimum value for the range. Notice that only the denominator is affected by #x#. In this case, we wish to have the greatest and lowest denominator possible to minimize and maximize #F#.
#cos2x in [color(red)(-1), color(blue)(1)]#
The greatest denominator is #3 - (color(red)(-1))# which gives a minimum value of:
#5/(3 - (color(red)(-1))) = 5/4#
The lowest denominator is #3 - (color(blue)1)# which gives a maximum value of:
#5/(3 - (color(blue)1)) = 5/2#
The range of #F# must therefore be #[5/4, 5/2]#.
