Definition 1.7.2. One-to-One Function.
A function \(f\) is said to be one-to-one if for every possible output (\(y\)-value) in the range of \(f\text{,}\) there is exactly one input (\(x\)-value) in the domain of \(f\text{.}\)
In other words, in a one-to-one function, each possible input is paired with exactly one output AND each possible output is paired with exactly one input.
Example 1.7.3.
The set \(\{ (0,-2), ~ (1, \boldsymbol{-1}), ~ (4, 0), ~ (9,\boldsymbol{-1}),~ (16,-3) \}\) is a function, but it is not one-to-one.
Example 1.7.4.
The set \(\{ (0,-2), ~ (1, -1), ~ (2, 0), ~ (3, 1),~ (4,2) \}\) is a function and is one-to-one.
Definition 1.7.10. Inverse Function.
If a function \(f\) is one-to-one, then the function has an inverse, \(f^{-1}\text{.}\)
Two functions \(f\) and \(f^{-1}\) are inverse functions if and only if both of the following are true:
The domain of a function \(f\) is the range of the inverse function \(f^{-1}\text{.}\)
The range of a function \(f\) is the domain of the inverse function \(f^{-1}\text{.}\)
Example 1.7.11.
If \(f\) is defined by the set
\begin{equation*}
\{ (0,-2), ~ (1, -1), ~ (2, 0), ~ (3, 1),~ (4,2) \},
\end{equation*}
then \(f^{-1}\) is the set
\begin{equation*}
\{ (-2,0), ~ (-1, 1), ~ (0, 2), ~ (1, 3),~ (2,4) \}.
\end{equation*}
The domain of \(f\) is \(\{ 0, ~ 1, ~ 2, ~ 3, ~ 4 \}\) and the range is \(\{ -2, \, -1, ~ 0, ~ 1,~ 2 \}\text{.}\)
The domain of \(f^{-1}\) is \(\{ -2, \, -1, ~ 0, ~ 1,~ 2 \}\) and the range is \(\{ 0, ~ 1, ~ 2, ~ 3, ~ 4 \}\text{.}\)
