Final answer:
The inverse function of f(x) = e^x is f^-1(x) = ln(x), with a domain of (0, ∞) and a range of all real numbers. It has no zeros as ln(x) does not equal zero for any positive x.
Step-by-step explanation:
When looking to find the inverse function of f(x) = e^x, we need to switch x and y and then solve for y. This process will tell us the inverse function f-1(x) and also help us determine its domain and range.
Starting with y = e^x, we switch x and y to get x = e^y. Taking the natural logarithm of both sides gives us ln(x) = y, which implies that the inverse function is f-1(x) = ln(x). The inverse function ln(x) only has real values for x > 0, meaning that its domain is (0, ∞) and its range is all real numbers, as the natural logarithm can take any real number as its output. The inverse function has no zeros because ln(x) never equals zero for any x in its domain.