How to define an opposite of a function. If for example I have the function F(x) = y how can use it to define the function f(y) = x
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Ilya Gazman
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This is called the inverse function. But such a function only exists if $f$ is bijective (if one restricts to the image of $f$ this then means one just needs $f$ to be injective). – Tobias Kildetoft Oct 16 '13 at 12:36
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@TobiasKildetoft Can you please refer me to a wiki page. And please submit it as an answer. – Ilya Gazman Oct 16 '13 at 12:39
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What you are seeking is called the inverse function $f^{-1}(x)$ of a bijective function $f(x)$. A function's inverse exists if and only if the function is bijective: if and only if it is both one-to-one and onto.
The process of finding the inverse depends on the bijective function. We can take something rather simple, for example $$f(x) = y = x^3$$
We first solve to express it as a function of $y$ ($f(y)): \sqrt[\large 3]{y} = \sqrt[\large 3]{x^3} = x$. Now we have $x = \sqrt[\large 3] y$. Then the inverse of the original function $f(x)$ is given by exchanging the positions of "x" and "y". $$f^{-1}(x) = \sqrt[\large 3]{x}$$
One can always check one's answer by evaluating the composite function to confirm that $$f^{-1}\Big(f(x)\Big) = f\left(f^{-1}(x)\right) = x$$
amWhy
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If $f: X\to Y$ is a function, then a function $g: Y\to X$ such that $f(x) = y$ if and only if $g(y) = x$ is called the inverse function of $f$ and is written $f^{-1}$.
If such a function is to exist, $f$ need to be bijective. It needs to be surjective for the function $g$ to be defined on all of $Y$, and it needs to be injective in order for the above to be well-defined.
More generally, $f$ is injective if and only if there is a function $g:Y\to X$ such that $f(x)=y$ implies $g(y) = x$, and $f$ is surjective if and only if there is a function $g: Y\to X$ such that $g(y) = x$ implies $f(x) = y$.
Tobias Kildetoft
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