Editing Inverse function (section)

== Real-world examples ==
* Let {{mvar|f}} be the function that converts a temperature in degrees [[Celsius]] to a temperature in degrees [[Fahrenheit]], <math display="block"> F = f(C) = \tfrac95 C + 32 ;</math> then its inverse function converts degrees Fahrenheit to degrees Celsius, <math display="block"> C = f^{-1}(F) = \tfrac59 (F - 32) ,</math><ref name=":1">{{Cite web|title=Inverse Functions|url=https://www.mathsisfun.com/sets/function-inverse.html|access-date=2020-09-08|website=www.mathsisfun.com}}</ref> since <math display="block">
\begin{align}
f^{-1} (f(C)) = {} & f^{-1}\left(  \tfrac95 C + 32 \right) = \tfrac59 \left( (\tfrac95 C + 32 ) - 32 \right) =  C, \\
& \text{for every value of } C, \text{ and } \\[6pt]
f\left(f^{-1}(F)\right) = {} & f\left(\tfrac59 (F - 32)\right) = \tfrac95 \left(\tfrac59 (F - 32)\right) + 32 = F, \\
& \text{for every value of } F.
\end{align}
</math>
* Suppose {{mvar|f}} assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has children born in the same year (for instance, twins or triplets, etc.) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example, <math display="block">\begin{align}
 f(\text{Allan})&=2005 , \quad & f(\text{Brad})&=2007 , \quad & f(\text{Cary})&=2001 \\
 f^{-1}(2005)&=\text{Allan} , \quad & f^{-1}(2007)&=\text{Brad} , \quad & f^{-1}(2001)&=\text{Cary}
\end{align}
</math>
* Let {{mvar|R}} be the function that leads to an {{mvar|x}} percentage rise of some quantity, and {{mvar|F}} be the function producing an {{mvar|x}} percentage fall. Applied to $100 with {{mvar|x}} = 10%, we find that applying the first function followed by the second does not restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are not inverses of each other.
* The formula to calculate the pH of a solution is {{math|1=pH = −log<sub>10</sub>[H<sup>+</sup>]}}. In many cases we need to find the concentration of acid from a pH measurement. The inverse function {{math|1=[H<sup>+</sup>] = 10<sup>−pH</sup>}} is used.