Fill in the blanks with the appropriate word, phrase, or...

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list. $$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$ If a one-to-one function has ordered pairs of the form $(x, y)$, the ____ function is a one-to-one function with ordered pairs of the form $(y, x)$.

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$
If a one-to-one function has ordered pairs of the form $(x, y)$, the ____ function is a one-to-one function with ordered pairs of the form $(y, x)$.

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Key Concepts

Domain and Range

The domain of a function is the complete set of all possible input values, while the range is the set of all outputs produced by the function. When finding an inverse function, these two sets swap roles—the range of the original function becomes the domain of the inverse, and the domain becomes the range.

Inverse Function

An inverse function is produced by reversing the roles of the inputs and outputs in a function. For every input-output pair (x, y) in the original function, the inverse function includes the pair (y, x). This concept only applies when the original function is one-to-one, ensuring that the inverse is properly defined as a function.

One-to-one Function

A one-to-one function, or an injective function, is one in which each unique element of the domain maps to a unique element in the range. This property is essential for a function to have an inverse, since the reversal process requires that each y-value corresponds to exactly one x-value.

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