Answer:
The domain of f^(-1)(x) is all real numbers.
Explanation:
To find the domain of the inverse function, f^(-1)(x), we first need to determine the domain of the original function, f(x), and then find the corresponding range of f(x), which will become the domain of its inverse.
The function f(x) = ln(2 + ln(x)) involves natural logarithms, and the domain of ln(x) is all positive real numbers. However, ln(x) is only defined when its argument (the number inside the logarithm) is greater than 0.
So, let's consider the domains step by step:
1. The inner logarithm ln(x) is defined when x > 0.
2. Adding 2 to ln(x) gives us 2 + ln(x), which is defined as long as ln(x) is defined, so it's also when x > 0.
3. Finally, taking the natural logarithm of 2 + ln(x) is defined for positive values of 2 + ln(x).
So, the domain of f(x) is all real numbers where x > 0.
Now, for the inverse function f^(-1)(x), the domain of f^(-1)(x) will be the range of f(x). In this case, the range of f(x) consists of all real numbers, since the natural logarithm can take any real number as input.
Therefore, the domain of f^(-1)(x) is all real numbers.