Which of the following functions from A={x ∈ R : −1≤x≤1} to itself are bijections?

Question:

Which of the following functions from $A=\{x \in R:-1 \leq x \leq 1\}$ to itself are bijections?

(a) $f(x)=|x|$

(b) $f(x)=\sin \frac{\pi x}{2}$

(c) $f(x)=\sin \frac{\pi^{2} x}{4}$

(d) None of these

Solution:

\text { (b) } f(x)=\sin \frac{\pi x}{2}

It is clear that $f(x)$ is one-one.

Range of $f=\left[\sin \frac{\pi(-1)}{2}, \sin \frac{\pi(1)}{2}\right]=\left[\sin \frac{-\pi}{2}, \sin \frac{\pi}{2}\right]=[-1,1]=A=$ Co domain of $f$

$\Rightarrow f$ is onto.

So, $f$ is a bijection.

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