what is the domain and range and horizontal asymptote of fx

what is the domain and range and horizontal asymptote of

f(x) = 5^-x -4

Solution

The domain of the function is all real numbers because the domain of the exponential function is all real numbers. The range can be determined by investigating the end behavior. As x approaches - infinity, the function itself approaches + infinity. As x approaches + infinity, the function itself approaches -4, but does not quite get there. Thus the range is all real numbers y such that y > -4.

The end behavior as x approaches + infinity (approaching -4 but never getting there) also shows us that there is a horizontal asymptote: namely the straight line y = -4.