Functions Test ...

If $$\displaystyle f(x)=\frac{1}{1-x},x\neq 0,1$$ then the graph of the function $$\displaystyle y=f\left \{ f(f(x)) \right \},x> 1,$$ is

If $$f$$ and $$g$$ are two functions such that  $$\displaystyle \left ( fg \right )\left ( x \right )=\left ( gf \right )\left ( x \right )$$ for all $$x$$. Then $$f $$ and $$g$$ may be defined as

If $$\displaystyle f(x)=x^{n},n\in N$$ and $$(gof)(x)=ng(x)$$ then $$g(x)$$ can be 

The domain of the function $$\displaystyle f\left ( x \right )= \sqrt{\sin ^{-1}\left ( 2x \right )+\frac{\pi }{6}}$$ for real $$x$$, is

If $$\displaystyle f\left ( x \right )=\left\{\begin{matrix}
x^{2}         x \geq 0\\
x              x < 0
\end{matrix}\right.$$
then $$\displaystyle (f o f)(x)$$ is given by

Let $$\displaystyle g(x)=1+x-[x]$$ and $$\displaystyle f(x)=\left\{\begin{matrix}{-1}\quad {x< 0} \\ {0} \quad {x=0}\\{1} \quad {x> 0} \end{matrix}\right.$$ Then for all  $$\displaystyle x, f\left \{ g\left ( x \right ) \right \}$$ is equal to 

Let $$\displaystyle f(x)=\frac{ax}{x+1}$$, where $$\displaystyle x\neq -1$$. Then for what value of $$\displaystyle a$$ is $$\displaystyle f( f(x))=x$$ always true

If $$\displaystyle f(y)=\frac{y}{\sqrt{1-y^2}}$$; $$\displaystyle g(y)=\frac{y}{\sqrt{1+y^2}}$$ then $$(fog)y$$ is equal to

If $$\displaystyle f(x)= (x-1)+(x+1)$$ and
$$\displaystyle g(x)= f\left \{ f(x) \right \}$$ then $$\displaystyle {g}'(3)$$

Let f(x)=tan x, x$$\displaystyle \epsilon \left [ -\frac{\pi }{2},\frac{\pi }{2} \right ]$$ and $$\displaystyle g\left (x  \right )=\sqrt{1-x^{2}}$$ Determine $$g o f(1)$$.