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 $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
From Graph of Sine Function this is apparent:
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 23$: Restriction of a Mapping