Inverse of Strictly Monotone Function

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Theorem

Let $f$ be a real function which is defined on $I \subseteq \R$.

Let $f$ be strictly monotone on $I$.

Let the image of $f$ be $J$.


Then $f$ always has an inverse function $f^{-1}$ and:

if $f$ is strictly increasing then so is $f^{-1}$
if $f$ is strictly decreasing then so is $f^{-1}$.


Proof

The function $f$ is a bijection from Strictly Monotone Real Function is Bijective.

Hence from Bijection iff Inverse is Bijection, $f^{-1}$ always exists and is also a bijection.


From the definition of strictly increasing:

$x < y \iff \map f x < \map f y$

Hence:

$\map {f^{-1} } x < \map {f^{-1} } y \iff \map {f^{-1} } {\map f x} < \map {f^{-1} } {\map f y}$

and so:

$\map {f^{-1} } x < \map {f^{-1} } y \iff x < y$


Similarly, from the definition of strictly decreasing:

$x < y \iff \map f x > \map f y$

Hence:

$\map {f^{-1} } x < \map {f^{-1} } y \iff \map {f^{-1} } {\map f x} > \map {f^{-1} } {\map f y}$

and so:

$\map {f^{-1} } x < \map {f^{-1} } y \iff x > y$

$\blacksquare$


Sources