Definition:Bijective Restriction

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Let $f: S \to T$ be a mapping which is not bijective.

A bijective restriction of $f$ is a restriction $f {\restriction_{S' \times T'} }: S' \to T'$ of $f$ such that $f {\restriction_{S' \times T'} }$ is a bijection.


Bijective Restrictions of $f \paren x = x^2 - 4 x + 5$

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x^2 - 4 x + 5$

The following real functions are bijective restrictions of $f$:

\(\ds f_1: \hointl \gets 2\) \(\to\) \(\ds \hointr 1 \to\)
\(\ds f_2: \hointr 2 \to\) \(\to\) \(\ds \hointr 1 \to\)