Question

$K\left(x\right)$ is a function such that $K\left(f\right(x\left)\right)=a+b+c+d$,
Where,
$a=\begin{cases}
0 & \text{ if f(x) is even} \\
-1 & \text{ if f(x) is odd} \\
2 & \text{ if f(x) is neither even nor odd}
\end{cases}$
$b=\begin{cases}
3 & \text{ if f(x) is periodic} \\
4 & \text{ if f(x) is aperiodic}
\end{cases}$
$c=\begin{cases}
5 & \text{ if f(x) is one one} \\
6 & \text{ if f(x) is many one}
\end{cases}$
$d=\begin{cases}
7 & \text{ if f(x) is onto} \\
8 & \text{ if f(x) is into}
\end{cases}$
$h:R\to R,h\left(x\right)=\left(\frac{e^{2x}+e^x+1}{e^{2x}-e^x+1}\right)$
On the basis of above information, answer the following questions.$K\left(\varphi \right(x\left)\right)$

• A. $15$
• B. $16$
• C. $17$
• D. $18$

Answer 1

The correct option is A $15$

$\varphi \left(x\right)\to (\frac{-\pi }{2},\frac{\pi }{2})$
Now we know that for the given domain, $\tan \left(x\right)$ is one-one, periodic, odd and an into function.
Hence $f\left(\varphi \right(x\left)\right)$
$=-1+3+5+8$
$=16-1$
$=15$