Question

The function $f:\left[-1/2,1/2,1/2\right]\to \left[-\pi /2,\pi /2\right]$ , defined by $f\left(x\right)={\sin }^{-1}\left(3x-4x^3\right)$, is
(a) bijection
(b) injection but not a surjection
(c) surjection but not an injection
(d) neither an injection nor a surjection

Answer 1

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

Injectivity:
Let x and y be two elements in the domain $\left[\frac{-1}{2},\frac{1}{2}\right]$ , such that
$$\begin{gathered} f\left(x\right)=f\left(y\right) \\ \Rightarrow 3{\sin }^{-1}x=3{\sin }^{-1}y \\ \Rightarrow {\sin }^{-1}x={\sin }^{-1}y \\ \Rightarrow x=y \end{gathered}$$
So, f is one-one.

Surjectivity:
Let y be any element in the co-domain, such that
$$\begin{gathered} f\left(x\right)=y \\ \Rightarrow 3{\sin }^{-1}\left(x\right)=y \\ \Rightarrow {\sin }^{-1}\left(x\right)=\frac{y}{3} \\ \Rightarrow x=\sin \frac{y}{3}\in \left[\frac{-1}{2},\frac{1}{2}\right] \end{gathered}$$
$\Rightarrow$f is onto.
$\Rightarrow$f is a bijection.
So, the answer is (a).