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A one-to-one function, also known as an injective function, is a type of function where every unique input is mapped to a unique output. This means that no two different inputs in the domain of the function can produce the same output in the range.
Identifying whether a function is one-to-one is crucial, especially when determining its inverse. If a function is not one-to-one, it doesn't have an inverse because multiple inputs could produce the same output, which would make reversing the process ambiguous.
To check if a function is one-to-one, we can use either the horizontal line test or the derivative method.

The horizontal line test is a visual way to determine if a function is one-to-one.
If every horizontal line intersects the graph of the function at most once, the function passes the test and is considered one-to-one.
For the function given in the exercise, we can see that it's a linear equation, specifically a straight line with a positive slope. A straight line with a positive or negative slope (and not horizontal) will never be intersected more than once by any horizontal line, thus passing the horizontal line test.

Another method to determine if a function is one-to-one is by using its derivative.
The derivative of a function provides information about the rate at which the function's value is changing.
For a function to be one-to-one, we need its derivative to either always be positive (indicating a strictly increasing function) or always be negative (indicating a strictly decreasing function).

In the given exercise, the function is linear:
\(f(x) = 3x + 2\).
The derivative of this function is
\(f'(x) = 3\),
which is a constant and positive value.
This tells us that the function is always increasing. Because the function is strictly increasing, it is guaranteed to be one-to-one.
We can then confidently proceed to find its inverse.