Find ƒ^-1(x), and give the domain and range. ƒ(x) = e^(x-5)

An inverse function reverses the effect of the original function. For a function f(x), its inverse f^-1(x) satisfies the condition f(f^-1(x)) = x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y. Understanding how to derive and interpret inverse functions is crucial for solving problems involving them.

Exponential functions are of the form f(x) = a * b^x, where a is a constant, b is the base, and x is the exponent. In this case, f(x) = e^(x-5) is an exponential function with base e, which is approximately 2.718. These functions are characterized by their rapid growth or decay and have specific properties regarding their domain and range, which are essential for understanding their behavior.

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). For the function f(x) = e^(x-5), the domain is all real numbers, as exponential functions are defined for every real number. The range, however, is limited to positive real numbers, reflecting the nature of exponential growth.