Question
Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R$, defined by $f(x) = sinx$
$f : R \rightarrow R$, defined by $f(x) = sinx$
✓
Answer
$f: R \rightarrow R$, given by $f(x)=\sin x$ Injective: Let $x, y \in R$ such that $f(x)=f(y) \Rightarrow \sin x=\sin y \Rightarrow x=n \pi+(-1)^n y$ $\Rightarrow x \neq y$
$\therefore f$ is not one-one.
Surjective: Let $y \in R$ be arbitrary such that
$f(x)=y$
$\Rightarrow \sin x=y \Rightarrow x=\sin ^{-1} y$ Now, for $y>1 \times \notin R$ (domain). $\therefore$ is not onto.
$\therefore f$ is not one-one.
Surjective: Let $y \in R$ be arbitrary such that
$f(x)=y$
$\Rightarrow \sin x=y \Rightarrow x=\sin ^{-1} y$ Now, for $y>1 \times \notin R$ (domain). $\therefore$ is not onto.
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