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)
