Relations and F...

$$K(x)$$ 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.$$K(\phi(x)) $$

Let $$f:{x, y, z}\rightarrow (a, b, c)$$ be a one-one function. It is known that only one of the following statements is true:

If $$f:R\rightarrow R$$ and $$g:R\rightarrow R$$ are given by $$f(x)=|x|$$ and $$g(x)=[x]$$ for each $$x\in R,$$ then $$\left\{ x\in R:g\left( f\left( x \right) \right) \le f\left( g\left( x \right) \right)  \right\} =$$

Let $$\displaystyle f:\left[-\frac{\pi}{3},\frac{2\pi}{3}\right]\rightarrow [0,4]$$ be a function defined by $$\displaystyle f(x)=\sqrt 3 \sin x-\cos x+2$$ then $$\displaystyle f^{-1}(x)$$ equals

Suppose $$\displaystyle  f\left ( x \right )=\left ( x+1 \right )^{2}$$ for $$\displaystyle x\geq -1$$. If $$\displaystyle g(x)$$ is the function whose graph is the reflection of the graph of $$\displaystyle f(x)$$ with respect to the line $$y=x $$, then $$\displaystyle g(x)$$ is equal to

If $$\displaystyle f\left ( x \right )= \frac{3x+2}{5x-3}$$ , then

If $$\displaystyle f\left ( x \right )+f\left ( \dfrac1x \right )=0,\; f(e)=1; g\left ( x \right )=f^{-1}\left ( x \right )$$ then $$\displaystyle g{}'\left ( x \right )$$ equals

If $$f:[1,\infty )\rightarrow [2,\infty )$$ is given by $$\displaystyle f\left( x \right)=x+\frac { 1 }{ x } ,$$ then $$f^{-1}(x)$$ equals

Let $$\displaystyle f\left ( x \right )=\frac{ax^{2}+2x+1}{2x^{2}-2x+1}$$, the value of $$a$$ for which $$\displaystyle f:R\rightarrow \left [ -1,2 \right ]$$ is onto , is

The total number of injective mappings from a set with $$m$$ elements to a set with $$n$$ elements, $$m \leq n $$ is