Functions Test ...

$$l= \lim_{x\rightarrow \alpha}\displaystyle \frac{f(x)}{x(x-\alpha)(x-2)}$$ is

The domain of the function $$\displaystyle y=\underbrace { \log _{ 10 } \log _{ 10 } ...\log _{ 10 } x }_{ \text{ n times } } $$ is

If $$\displaystyle f \left ( x \right ) = px + q$$ and $$\displaystyle f \left ( f\left ( f\left ( x \right ) \right ) \right ) = 8x + 21$$, where $$p$$ and $$q$$ are real numbers, the $$ p + q$$ equals

The value of $$(a + b)$$ is equal to

The domain of the function $$\displaystyle f\left( x \right)=\sin ^{ -1 }{ \left\{ \log _{ 2 }{ \left( \frac { 1 }{ 2 } { x }^{ 2 } \right)  }  \right\}  } $$ is

$$K(x)$$ 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:

The domain of function $$\displaystyle f(x)=\sqrt{x-\sqrt{1-x^{2}}}$$ is

The domain of the function $$\displaystyle f(x)=\sqrt{1-\sqrt{1-\sqrt{1-x^{2}}}}$$ is

Let $$\displaystyle f(x)=\sin^{2}\dfrac{x}{2}+\cos^{2}\frac{x}{2}$$ and $$g(x)=\sec^{2}x-\tan^{2}x.$$ The two functions are equal over the set