Question

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

• A. both one-one and onto
• B. neither one-one nor onto
• C. onto but not one-one
• D. one-one but not onto

Answer 1

Answer: (A)

Since $\sin ^{-1}\left(3 x-4 x^3\right)=3 \sin ^{-1} x \in[-\pi / 2, \pi / 2]$ i.e., $\sin ^{-1} x \in[-\pi / 6, \pi / 6]$
or $x \in[-1 / 2,1 / 2]$ so $f$ is onto.
Also $f^{\prime}(x)=\frac{3}{\sqrt{1-x^2}} >0$ for $-1 / 2< x< 1 / 2$.
Therefore, $f$ increases on $[-1 / 2,1 / 2]$ and hence $f$ is one-one.