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Question
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Options
\[f\left( x \right) = |x|\]
\[f\left( x \right) = \sin\frac{\pi x}{2}\]
\[f\left( x \right) = \sin\frac{\pi x}{4}\]
None of these
Solution
\[f\left( x \right) = \sin\frac{\pi x}{2}\]
It is clear that f(x) is one-one.
\[\text{Range of f} = \left[ \sin\frac{\pi\left( - 1 \right)}{2}, \sin\frac{\pi\left( 1 \right)}{2} \right] = \left[ \sin \frac{- \pi}{2}, \sin\frac{\pi}{2} \right] = \left[ - 1, 1 \right] = A = \text{Co domain of f}\]
⇒ f is onto.
So, f is a bijection.
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