Short Answer 
The domain of function f is x > 0, with a range of all real numbers. Its inverse function has a domain of all real numbers and a range of y > 0, illustrating a symmetry where the domain of f corresponds to the range of its inverse and vice versa.
Step 1: Identify the Domain and Range of Function f
For function f, the domain is defined as all values of x that are greater than 0, which can be expressed as x > 0. The range of function f includes all real numbers, meaning that f can output any value along the number line, covering all possibilities.
Step 2: Determine the Domain and Range of the Inverse Function
The inverse function of f has its own domain and range. The domain of the inverse function is all real numbers, indicating that it can accept any value as input. Conversely, the range of the inverse function is defined as all values of y that are greater than 0, denoted as y > 0.
Step 3: Relationship Between the Domains and Ranges
It’s important to understand the relationship between the domains and ranges of both functions. Specifically, the domain of function f corresponds to the range of its inverse function, and the range of function f matches the domain of the inverse function. This symmetry plays a critical role in the characteristics of inverse functions:
- The domain of f (x > 0) is equal to the range of the inverse.
- The range of f (all real numbers) is equal to the domain of the inverse.