Loterre: Mathématiques: inverse function

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inverse function

In mathematics, the inverse function of a function $f$ (also called the inverse of $f$ ) is a function that undoes the operation of $f$ . The inverse of $f$ exists if and only if $f$ is bijective, and if it exists, is denoted by ${\\displaystyle f^{-1}.}$
For a function ${\\displaystyle f\\colon X\ o Y}$ , its inverse ${\\displaystyle f^{-1}\\colon Y\ o X}$ admits an explicit description: it sends each element ${\\displaystyle y\\in Y}$ to the unique element ${\\displaystyle x\\in X}$ such that $f (x) = y$ .
As an example, consider the real-valued function of a real variable given by $f (x) = 5 x - 7$ . One can think of $f$ 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 $f$ is the function ${\\displaystyle f^{-1}\\colon \\mathbb {R} \ o \\mathbb {R} }$ defined by ${\\displaystyle f^{-1}(y)={\ rac {y+7}{5}}.}$