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Algebraic manipulation is an essential tool in mathematics, particularly when finding the inverse of functions. It involves rearranging equations and expressions using arithmetic operations and properties of equality to solve for a particular variable. In the context of inverse functions, algebraic manipulation is used to 'undo' the original function.

For instance, to find the inverse of a given function like the one in the exercise, you start by expressing the function with a variable, usually denoted by y. Swapping the roles of x (typically the input) and y (the output), and then solving for y involves a series of algebraic steps. This includes multiplying both sides of an equation to eliminate fractions or adding terms to both sides to get the variable of interest on its own.

The original exercise demonstrates a clear cut application of algebraic manipulation by first substituting f(x) with y, swapping x and y, then multiplying and dividing to isolate y. By following these algebraic steps methodically, finding the inverse can be made accessible even to those new to the concept.

A one-to-one function, also known as an injective function, is a type of function in which every element of the function's range corresponds precisely to one element of its domain. Simply put, no two different inputs will lead to the same output in a one-to-one function.

This property is crucial when considering inverse functions because only one-to-one functions have inverses that are also functions. If a function were not one-to-one, its inverse would assign multiple outputs to a single input, which defies the definition of a function.

To determine if a function is one-to-one, you can use the Horizontal Line Test. If every horizontal line intersects the graph of the function at most once, the function is one-to-one. In the exercise, the function $$ f(x) = \frac{1}{x} $$ is injective since for every x-value, there exists only one unique y-value, thus meeting the one-to-one criterion and allowing for the existence of an inverse function.

Inverse notation is a mathematical convention used to denote the inverse of a function. Inverse functions are denoted by an exponent of -1, written as $$ f^{-1}(x) $$. It's important to note that this notation does not imply an exponent in the arithmetic sense; rather, it signifies that the function $$ f^{-1}(x) $$ will 'undo' the work of the original function $$ f(x) $$.

When you apply the inverse function to the output of the original function, you should get back the initial input, which means that $$ f^{-1}(f(x)) = x $$ for all x in the domain of f. Similarly, applying the original function to the output of the inverse function gives $$ f(f^{-1}(x)) = x $$.

In the step-by-step solution of the exercise, once we have isolated y in terms of x, the inverse function is expressed using the inverse notation: $$ f^{-1}(x) $$. It suggests a reversal of the process described by the original function. In our specific case, the interesting aspect is that the inverse function of $$ f(x) = \frac{1}{x} $$ is the function itself, denoted by $$ f^{-1}(x) = \frac{1}{x} $$.