The function f : [−1/2, 1/2, 1/2]→[−π/2, π/2] , defined by


The function $f:[-1 / 2,1 / 2,1 / 2] \rightarrow[-\pi / 2, \pi / 2]$, defined by $f(x)=\sin ^{-1}\left(3 x-4 x^{3}\right)$, is

(a) bijection
(b) injection but not a surjection
(c) surjection but not an injection
(d) neither an injection nor a surjection


$f(x)=\sin ^{-1}\left(3 x-4 x^{3}\right)$

$\Rightarrow f(x)=3 \sin ^{-1} x$


Let $x$ and $y$ be two elements in the domain $\left[\frac{-1}{2}, \frac{1}{2}\right]$, such that


$\Rightarrow 3 \sin ^{-1} x=3 \sin ^{-1} y$

$\Rightarrow \sin ^{-1} x=\sin ^{-1} y$

$\Rightarrow x=y$

So. $f$ is one-one.

Let y be any element in the co-domain, such that


$\Rightarrow 3 \sin ^{-1}(x)=y$

$\Rightarrow \sin ^{-1}(x)=\frac{y}{3}$

$\Rightarrow x=\sin \frac{y}{3} \in\left[\frac{-1}{2}, \frac{1}{2}\right]$

$\Rightarrow f$ is onto.

$\Rightarrow f$ is a bijection.

So, the answer is (a).

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