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.