The domain of a function is the range of real numbers the variable X can take such that #h(x)# is real. The range is the set of all values which #h(x)# can take when #x# is assigned a value in the domain.
Here we have a polynomial involving the subtraction of an exponential. The variable is really only involved in the #-4^x# term, so we'll work with that.
There are three primary values to check here: #x<-a, x=0, x>a#, where #a# is some real number. #4^0# is simply 1, so #0# is in the domain. Plugging in various positive and negative integers, one determines that #4^x# yields a real result for any such integer. Thus, our domain is all real numbers, here represented by #[-oo,oo]#
How about the range? Well, first note the range of the second part of the expression, #4^x#. If one puts in a large positive value, one gets a large positive output; putting in 0 yields 1; and putting in a 'large' negative value yields a value very close to 0. Thus, the range of #4^x# is #(0,oo)#. If we place these values into our initial equation, we learn that the lower bound is #-oo# (#6-4^x# goes to #-oo# as x goes to #oo#), and the upper bound is 6 (#h(x))# goes to #6# as #x->-oo#)
Thus, we arrive at the following conclusions.
Domain: #(-oo,oo)#
Range: #(-oo, 6)#
