Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = \sin x$

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Answer

$f : R \rightarrow R,$ given by $f(x) = \sin x$
Injective: Let $\text{x, y}\in\text{R}$
such that $f(x) = f(y)$
$\Rightarrow \sin x = \sin y$
$\Rightarrow\ \text{x}=\text{n}\pi+(-1)^{\text{n}}\text{y}$
$\Rightarrow\ \text{x}\neq\text{y}$
$\therefore f $ is not one-one.
Surjective: Let $\text{y}\in\text{R}$ be arbitrary such that
$f(x) = y$
$\Rightarrow \sin x = y \Rightarrow x = \sin^{-1}y$
Now, for $\text{y}>1\times\notin\text{R}$ (domain)
$\therefore$ f is not onto.

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