Final answer:
The domain and range for each inverse function are determined by the properties of logarithmic and exponential functions, and the specific expressions given.
Step-by-step explanation:
To determine the domain and range of inverse functions, you need to understand the behavior of the function in question. Here are the answers to the problems given:
- Domain and range of f^(-1)(x) = ln(x): The domain of the natural log function (ln) is (0, +infinity) because you cannot take the logarithm of zero or a negative number. The range is (-infinity, +infinity) because the logarithm can output any real number.
- Domain and range of g^(-1)(x) = sqrt(4 - x^2): The domain is the set of values for which the expression inside the square root is non-negative, which is [-2, 2]. The range of the inverse function, which is the original domain of g(x), depends on g(x) itself but assuming g(x) is the main branch of the square function, the range would be [0, +infinity).
- Domain and range of h^(-1)(x) = e^x: The domain is (-infinity, +infinity) because the exponential function is defined for all real numbers. The range is (0, +infinity) because e^x is always positive.
- Domain and range of k^(-1)(x) = 1/(x - 2): The domain cannot include the value that makes the denominator zero, so it is all real numbers except x=2. The range is all real numbers except y=0 because the function will never equal zero.
When calculating these values on a calculator, use the relevant function buttons to explore these behaviors for inverse functions such as the natural log (ln) and the exponential function (e^x).