Question

If the following functions have both domain and co-domain as $[-1,1]$, then select those which are not bijective?

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

Answer 1

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

$\text{sgn}\left(x\right)x^3$

$f\left(x\right)=\sin \left({\sin }^{-1}x\right)=x\forall x\in [-1,1]$, which is one-one and onto.
$f\left(x\right)=\frac{2}{\pi }\sin \left({\sin }^{-1}x\right)=\frac{2}{\pi }x$
The range of the function for $x\in [-1,1]$ is $[-\frac{2}{\pi },\frac{2}{\pi }]$, which is a subset of $[-1,1]$.
Hence, the function is one-one but not onto and, hence, is not bijective.

$f\left(x\right)=\text{sgn}\left(x\right)\ln \left(e^x\right)=\left(\text{sgn}\right(x\left)\right)x=\{\begin{array}{cc}x,& x>0\\ -x,& x<0\\ 0,& x=0\end{array}$

This function has range $[0,1]$ which is a subset of $[-1,1]$.
Hence, the function is into. Also, the function is many-one.

$f\left(x\right)=x^3\text{sgn}\left(x\right)=\{\begin{array}{cc}{x}^3,& x>0\\ -x^3,& x<0\\ 0,& x=0\end{array}$
which is many-one and into.