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In mathematics, a function is considered *one-to-one* if each output is determined by exactly one unique input. This means no two different input values (let's call them x-values) will produce the same output (or y-value). This concept is crucial when determining if an inverse function exists.

To find if a function has an inverse, it's essential to determine if that function is one-to-one. A graphical way to test this is by using the horizontal line test. If no horizontal line crosses the graph of the function more than once, then the function is one-to-one.

In the provided exercise, once we've found the inverse function, we evaluate it to check if each y-value leads to a unique x-value. As noted in the solution, there’s a critical point where the inverse may fail: when the denominator becomes zero (in our example, when y = 1). If an inverse function tries to map multiple x-values to the same y-value, it is not one-to-one, and thus, doesn’t have an inverse.

Function composition is a way to combine two functions into a single new function. Imagine you have two functions, f(x) and g(x). If you want to apply g first and then f, you are performing function composition, denoted as f(g(x)). This can also help verify if one function is the inverse of another.

The relationship between a function and its inverse is best highlighted through composition. If f is a function and g is supposed to be its inverse, two key equalities must hold true:

• f(g(x)) = x for every x in the domain of g
• g(f(x)) = x for every x in the domain of f

Checking these conditions helps ensure that a function is indeed the inverse of another, thereby confirming they're both one-to-one relationships.

Solving equations is at the heart of finding an inverse function. To find the inverse, you need to manipulate the original equation and solve for x in terms of y.

Think of the given function y=f(x). To find its inverse, your goal is to express x in terms of y, effectively flipping the roles of x and y. This is done by undoing the operations that were initially done to x. Here’s a structured way to approach solving for an inverse:

• Start with the original equation and exchange x and y.
• Solve the resulting equation for y as your new variable, isolating it on one side of the equation.

In the provided solution, the equation y = \(\frac{x}{x-1}\) was restructured to express x in terms of y, leading to x=\(\frac{y}{y-1}\). This step-by-step manipulation is what gives us the inverse function when possible.