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. $$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$ For any one-to-one function $f(x)$ and its inverse $f^{-1}(x)$, the range of $f(x)$ is the _____ and $\left(f^{-1} \circ f\right)(x)=$ ____.

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$$\begin{array}{lllll}\text { inverse } & \text { domain } & \text { horizontal } & x & f(x) \\\text { vertical } & \text { range } & \text { one-to-one } & \text { composition } & y\end{array}.$$
For any one-to-one function $f(x)$ and its inverse $f^{-1}(x)$, the range of $f(x)$ is the _____ and $\left(f^{-1} \circ f\right)(x)=$ ____.

Key Concepts

Inverse Function

An inverse function essentially reverses the action of the original function. When a function f is one-to-one, its inverse f?¹ exists such that for every output value of f, f?¹ returns the corresponding input value. The roles of the domain and range are swapped between the function and its inverse.

One-to-One Function

A one-to-one function is one where each element in the domain maps to a unique element in the range, ensuring that no two distinct inputs produce the same output. This uniqueness is essential for the existence of an inverse function, because it guarantees that the reverse mapping is well-defined.

Domain

The domain of a function is the set of all possible inputs for which the function is defined. In the context of inverse functions, the domain of the original function becomes the range of its inverse, reflecting the reversal of roles between inputs and outputs.

Range

The range of a function is the set of all possible outputs that the function can produce. For a one-to-one function, the range of f is exactly the set of inputs for the inverse function, reinforcing the idea that the output of f is the input for f?¹.

Function Composition

Function composition involves applying one function to the result of another, denoted as (g ? f)(x) = g(f(x)). In the study of inverse functions, composing a function with its inverse yields the identity function on the domain of the original function.

Identity Function

The identity function is a function that returns its input unchanged, typically expressed as I(x) = x. When a function is composed with its inverse, the result is the identity function, thereby confirming the correctness of the inverse relationship.

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