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A one-to-one function is a special type of function where every element in the domain corresponds to a unique element in the range. This means that no two different inputs (or "x" values) produce the same output (or "y" value).

• In mathematical terms, for a function to be one-to-one, if \(f(a) = f(b)\), then it must follow that \(a = b\).
• One-to-one functions are essential for finding inverses. If a function is not one-to-one, we cannot uniquely determine an inverse function.

By ensuring that each input has a unique output, one-to-one functions allow us to reverse the operation effectively. In simpler terms, if you know the output, you can trace it back to a single input with certainty. This is why in the given exercise, the function being identified as one-to-one is the core first step in finding its inverse.

The domain and range of a function describe the set of allowable inputs and attainable outputs, respectively. This is a fundamental aspect of understanding functions.

• The domain of a function is the set of all possible input values. For \(f(x) = 10 - \frac{1}{5x}\), the domain includes all real numbers except zero, since division by zero is undefined.
• The range of a function is the set of all possible output values. For the function given, it can approach, but not actually reach, 10, leading to a range of \((-fty, 10) \cup (10, \infty)\).

When finding the inverse of a function, the domain and range swap roles. The range of the original function becomes the domain of the inverse, and vice versa. This emphasizes why understanding these sets is crucial when working with inverses.

Solving equations is a fundamental part of finding inverse functions. The process usually involves algebraic manipulation to isolate the variable of interest. Here’s how you do it, step by step:1. **Switch Variables**: When finding the inverse, swap the roles of \(x\) and \(f(x)\). If \(f(x) = y\), write the equation as \(f(y) = x\).2. **Isolate \(y\)**: Use algebraic steps to solve for \(y\). This might involve adding, subtracting, multiplying, dividing, or even factoring, depending on the function.For example, in the given exercise:- Start with \(x = 10 - \frac{1}{5y}\).- Add \(\frac{1}{5y}\) to both sides to get \(\frac{1}{5y} = 10 - x\).- Finally, solve for \(y\) to get \(y = \frac{1}{5(10-x)}\).This manipulation helps us express \(y\) in terms of \(x\), taking us a step closer to identifying the inverse function.

Function notation is a way of writing functions that allows clear communication of the input-output relationship. It uses symbols like \(f(x)\) to represent the output of the function with respect to a given input \(x\).

• Pay attention to the "\(f^{-1}(x)\)" notation, which specifically signifies the inverse of the function \(f\).
• The inverse function \(f^{-1}(x)\) reverses the effect of the original function \(f(x)\), mapping outputs back to their original inputs.

Using function notation is not just a formal way of writing, but also aids in separating different processes mathematically. In problem-solving, moving from \(f(x)\) to \(f^{-1}(x)\) utilizing proper notation ensures clarity and precision in presenting ideas. This concise form is especially useful in calculus and other higher-level mathematics where operations on functions are frequent.