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One-to-one functions are special types of functions where every input has a unique output, and every output is connected to only one input. This means that the function never takes the same value twice. If you imagine the function as a machine, it works like a precise lock that only one specific key can open. This is why these functions are also called "injective functions."

• If \(f(a) = f(b)\), this implies that \(a = b\).
• No two distinct inputs point to the same output.

The one-to-one property is crucial for finding the inverse of a function. If a function is not one-to-one, it does not have a properly defined inverse. In the original exercise, for both parts \(a\) and \(b\), this property helps us directly relate the function \(f\) to its inverse \(f^{-1}\). In part (a), knowing \(f(5) = 18\) allows us to conclude \(f^{-1}(18) = 5\). Similarly, in part (b), \(f^{-1}(4) = 2\) leads to \(f(2) = 4\). This interchangeability between \(f\) and \(f^{-1}\) is the hallmark of a one-to-one function.

Function notation helps us to express mathematical functions clearly and concisely. It is like using a precise language to describe the relationship between variables. Typically, we use \(f\) to denote a function, and \(f(x)\) to denote the output of \(f\) when the input is \(x\). For inverse functions, we use \(f^{-1}(x)\). This does not mean \(1/f(x)\) - remember it's notation for the inverse function!

• \(f(a) = b\): Here, \(b\) is the output of the function \(f\) for the input \(a\).
• \(f^{-1}(b) = a\): This tells us that \(a\) is the input that gives an output of \(b\) in the function \(f\).

In the exercise, using proper function notation makes it easy to switch between the function and its inverse. When it says \(f(5) = 18\), it tells us the input-output relationship directly. For the inverse \(f^{-1}(18) = 5\) follows clearly, showing how function notation provides clarity and prevents confusion in solving the problem.

Understanding function properties is essential in solving problems related to functions and their inverses. Some of the crucial properties include the domain and range, and for one-to-one functions, specifically, they allow us to determine the inverse function reliably.

• Domain and Range: For every function, the domain is the set of all possible inputs, while the range is the set of all possible outputs.
• Inverses: A function \(f\) only has an inverse if it is one-to-one. The inverse function \(f^{-1}\) 'reverses' the mapping of \(f\).
• Reversibility: The equation \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) must hold true.

In the exercise, these properties make navigating between \(f\) and \(f^{-1}\) seamless. Since we've identified \(f\) as one-to-one, this ensures that the inverse function exists and can be used without ambiguity. This enables straightforward solutions for parts \(a\) and \(b\) given the properties of functions, allowing clear and accurate determination of \(f^{-1}(18)\) and \(f(2)\).