Restriction/Mapping/Examples/Bijective Restriction of Real Sine Function

Example of Restriction of Mapping
Let $f: \R \to \R$ be the mapping defined as:
 * $\forall x \in \R: f \paren x = \sin x$

Then a bijective restriction $g$ of $f$ can be defined as:


 * $g: S \to T: \forall x \in S: g \paren x = \sin x$

where:
 * $S = \closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
 * $T = \closedint {-1} 1$

Proof
We note that $f$ is neither injective nor surjective:

From Shape of Sine Function:

From Graph of Sine Function this is apparent: