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

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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:

The sine function is:

$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
$(3): \quad$ concave on the interval $\closedint 0 \pi$
$(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$


From Graph of Sine Function this is apparent:

Sine.png


$\blacksquare$


Sources