Shoe Sizes The table shows men's shoe sizes in the United...

Shoe Sizes The table shows men's shoe sizes in the United States and the corresponding European shoe sizes. Let $y=f(x)$ represent the function that gives the men's European shoe size in terms of $x,$ the men's U.S. size. $$ \begin{array}{|c|c|}\hline \text { Men's U.S. } & {\text { Men's European }} \\ \hline \text { shoe size } & {\text { shoe size }} \\ \hline 8 & {41} \\ {9} & {42} \\ {10} & {43} \\ {11} & {45} \\ {12} & {46} \\ {13} & {47} \\ \hline\end{array} $$ (a) Is $f$ one-to-one? Explain. (b) Find $f(11)$ (c) Find $f^{-1}(43),$ if possible. (d) Find $f\left(f^{-1}(41)\right)$ (e) Find $f^{-1}(f(13))$

Shoe Sizes The table shows men's shoe sizes in the United States and the corresponding European shoe sizes. Let $y=f(x)$ represent the function that gives the men's European shoe size in terms of $x,$ the men's U.S. size.
$$
\begin{array}{|c|c|}\hline \text { Men's U.S. } & {\text { Men's European }} \\ \hline \text { shoe size } & {\text { shoe size }} \\ \hline 8 & {41} \\ {9} & {42} \\ {10} & {43} \\ {11} & {45} \\ {12} & {46} \\ {13} & {47} \\ \hline\end{array}
$$
(a) Is $f$ one-to-one? Explain.
(b) Find $f(11)$
(c) Find $f^{-1}(43),$ if possible.
(d) Find $f\left(f^{-1}(41)\right)$
(e) Find $f^{-1}(f(13))$

Key Concepts

Composition of functions and their inverses

The composition of a function and its inverse, expressed as f(f?¹(x)) or f?¹(f(x)), yields the original value x within the appropriate domain. This property demonstrates that the inverse function effectively 'undoes' the action of the original function, confirming the relationship between the function and its inverse.

Domain and Range

The domain of a function is the set of all possible inputs, while the range is the set of all outputs produced by those inputs. Understanding domain and range is essential when analyzing functions, especially when considering whether an inverse function exists, as the inverse effectively switches these sets.

Inverse functions

An inverse function reverses the mapping of the original function; if f(x) = y, then the inverse function f?¹(y) = x. The existence of an inverse depends on the function being one-to-one, and the inverse function swaps the roles of inputs and outputs.

Function evaluation

Function evaluation involves substituting a specific value from the domain into the function to obtain the corresponding output. This process is used to determine f(x) for a given x by referencing the definition or rule of the function, as demonstrated by looking up a value in a table or computing based on a formula.

One-to-one function

A one-to-one function, or injective function, is one in which each input corresponds to a unique output, ensuring that no two different inputs map to the same output. This property is crucial when determining if a function can be inverted, as an inverse function exists only for one-to-one functions.

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