The logarithmic function \( f(x) = \ln(x) \) has a domain of \( x > 0 \), meaning it is only defined for positive values of \( x \). The corresponding exponential function \( g(x) = e^x \) is defined for all real numbers.
Specifically, the true statements about \( g(x) = e^x \) are:
- Domain of \( g(x) \): The domain of \( g(x) = e^x \) is all real numbers, or \( (-\infty, \infty) \).
- Range of \( g(x) \): The range of \( g(x) = e^x \) is \( (0, \infty) \), meaning it takes on all positive values.
- Inverse Relationship: The functions \( g(x) \) and \( f(x) \) are inverses of each other, meaning that \( g(f(x)) = x \) for \( x > 0 \) and \( f(g(x)) = x \) for all real \( x \).
Thus, while the logarithmic function \( f(x) = \ln(x) \) only accepts positive inputs, the exponential function \( g(x) = e^x \) can accept any real number as input.