Method 1
Consider #F(-x)#,
if #F(-x)=F(x)# for all #x# in the domain of #F#, then #F# is even
if #F(-x)=-F(x)# for all #x# in the domain of #F#, then #F# is odd
if #F(-x)# is not always #F(x)# and not always #-F(x)#, then #F# is neither even nor odd.
#F(x)=(2x)/absx#
#F(-x) = (2(-x))/abs(-x) = (-2x)/absx# #" "# #" "# (Note: #abs(-x)=absx#).
So, #F(-x)=-F(x)# for all #x# in the domain of #F#, and, therefore, #F# is odd.
Method 2
Simplify #F(x)=(2x)/absx# using the definition of absolute value.
#absx = {(" "x," if",x >= 0),(-x," if",x<0) :}#
Note that #F# is not defined at #x=0#, so we get:
#F(x) = {((2x)/x," if",x > 0),((2x)/(-x)," if",x<0) :}#.
Simplifying, gets us:
#F(x) = {(" "2," if",x > 0),(-2," if",x<0) :}#
At this point it is clear that #F(-x) = -F(x)# for all #x# in the domain of #F#. (Because for every #x != 0#, we have #-x# is on the oppposite side of #0#.)