Question

In the following functions defined from $[-1,1]$ to $[-1,1]$, then functions which are not bijective are

• A. $\sin \left({\sin }^{-1}x\right)$
• B. $\frac{2}{\pi }{\sin }^{-1}\left(\sin x\right)$
• C. $\left(sgnx\right)ln{e}^x$
• D. $x^{3}sgnx$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\frac{2}{\pi }{\sin }^{-1}\left(\sin x\right)$
C $\left(sgnx\right)ln{e}^x$
D $x^{3}sgnx$

$sin{\sin }^{-1}x=x$ hence it is bijective.

$\frac{2}{\pi }{\sin }^{-1}sinx$= $\frac{2x}{\pi }$. Hence it is one one but not onto, hence not bijective

$sgn\left(x\right).x=\frac{\left|x\right|}{x}.x=|x|,x\ne$0
Hence it is not one one or onto, hence not bijective

$x^{3}sgnx=x^3\frac{|x|}{x}=x^2|x|={|x|}^3,x\ne 0$. Hence function is not one one or onto, hence not bijective.