Which of the following represents the inverse of a function f(x)?

The representation of the inverse of a function f(x) is denoted as f^-1(x). This notation indicates a function that, when composed with the original function f(x), yields the identity function. In simpler terms, if you take an output from f(x) and insert it into f^-1(x), you will retrieve the original input value.

To illustrate, if f(a) = b, then by applying the inverse function, we have f^-1(b) = a. This duality is essential in understanding how functions relate to their inverses. The concept of an inverse function revolves around reversing the actions of the original function.

In contrast, the other choices represent different mathematical operations. The expression 1/f(x) denotes the reciprocal of the function's output, not an inverse. The term f(x + 1) indicates a shift in the input of the function, while f(-x) signifies reflection over the y-axis. None of these operations reciprocate the outputs and inputs of the original function like the inverse does, which is what distinguishes f^-1(x) as the correct choice.