#f(x)=x^2-2x-3# is defined for all Real values of #x#

therefore the Domain of #f(x)# covers all Real values (i.e. #x in RR#)

#x^2-2x-3# can be written in vertex form as #(x-color(red)1)^2+color(blue)((-4))# with vertex at #(color(red)1,color(blue)(-4))#

Since the (implied) coefficient of #x^2# (namely #1#) is positive, the vertex is a minimum

and #color(blue)((-4))# is a minimum value for #f(x)#;

#f(x)# increases without bound (i.e. approaches #color(magenta)(+oo)#) as #xrarr +-oo#

so #f(x)# has a Range of #[color(blue)(-4),color(magenta)(+oo))#