Editing Inverse function (section)

{{short description|Mathematical concept}}
{{Distinguish|multiplicative inverse|additive inverse}}
{{use dmy dates|date=August 2020|cs1-dates=y}}
[[Image:Inverse Function.png|thumb|right|A function {{mvar|f}} and its inverse {{math|''f''<sup> −1</sup>}}. Because {{mvar|f}} maps {{mvar|a}} to 3, the inverse {{math|''f''<sup> −1</sup>}} maps 3 back to {{mvar|a}}.]]
{{Functions}}
In [[mathematics]], the '''inverse function''' of a [[Function (mathematics)|function]] {{Mvar|f}} (also called the '''inverse''' of {{Mvar|f}}) is a [[function (mathematics)|function]] that undoes the operation of {{Mvar|f}}. The inverse of {{Mvar|f}} exists [[if and only if]] {{Mvar|f}} is [[Bijection|bijective]], and if it exists, is denoted by <math>f^{-1} .</math>

For a function <math>f\colon X\to Y</math>, its inverse <math>f^{-1}\colon Y\to X</math> admits an explicit description:  it sends each element <math>y\in Y</math> to the unique element <math>x\in X</math> such that {{Math|1=''f''(''x'') = ''y''}}.

As an example, consider the [[Real number|real-valued]] function of a real variable given by {{math|1=''f''(''x'') = 5''x'' − 7}}. One can think of {{Mvar|f}} as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of {{Mvar|f}} is the function <math>f^{-1}\colon \R\to\R</math> defined by <math>f^{-1}(y) = \frac{y+7}{5} .</math>