Question

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