Which of the following functions from $A=\{x:-1 \leq x \leq 1\}$ to itself are bijections?
(a) $f(x)=\frac{x}{2}$
(b) $g(x)=\sin \left(\frac{\pi x}{2}\right)$
(c) $h(x)=|x|$
(d) $k(x)=x^{2}$
(a) Range of $f=\left[\frac{-1}{2}, \frac{1}{2}\right] \neq A$
So, $f$ is not a bijection.
(b) Range $=\left[\sin \left(\frac{-\pi}{2}\right), \sin \left(\frac{\pi}{2}\right)\right]=[-1,1]=A$
So, $g$ is a bijection.
(c) $h(-1)=|-1|=1$
and $h(1)=|1|=1$
$\Rightarrow-1$ and 1 have the same images
So, $h$ is not a bijection.
(d) $k(-1)=(-1)^{2}=1$
and $k(1)=(1)^{2}=1$
$\Rightarrow-1$ and 1 have the same images
So, $k$ is not a bijection.
So, the answer is (b).
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