Question

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

A
15
B
16
C
17
D
18

Solution

## The correct option is B $$16$$$$g(x)=\sin(x)$$Hence $$g(-x)=-g(x)$$, therefore an odd function.It is periodic, with a period of $$\pi$$$$sin(x)$$ is an onto function only for a co-domain of $$\left[\dfrac{-\pi}{2},\dfrac{\pi}{2}\right]$$ since its range is $$[-1,1]$$Thus for $$R \rightarrow R$$ or a co-domain, R, it is an into function.Also $$\sin(x)$$ is a many on one function since$$\sin(x_{1})=\sin(x_{2})$$Implies$$x_{1}=(2n+1)\pi-x_{2}$$ where $$n\in W$$.Hence$$k(g(x))=-1+3+6+8$$$$=16$$Maths

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