Functions Test ...

The domain of $${f}({x})=\sqrt{2-\log_{3}(x-1)}$$  is

The domain of the function defined by $$ \mathrm{f}({x})=^{(7-x)}P_{(x-3)}$$ is

Let $$S$$ be set of all rational numbers. The functions $$f:R\rightarrow R,\ g:R\rightarrow R$$ are defined as 
$$f(x)=\begin{cases}
0, & x \in S \\ 
1, & x \notin S
\end{cases}$$
$$g(x)=\begin{cases}
-1 & x\in S \\ 
 0 & x\notin S
\end{cases}$$
then, $$(fog) (\pi)+(gof)(e)=$$

If $$n\geq 1$$ is any integer, $$\mathrm{d}(n)$$ denotes the number of positive factors of $$n$$, then for any prime number $$\mathrm{p},\ \mathrm{d}(\mathrm{d}(\mathrm{d}(\mathrm{p}^{7})))=$$

The domain of $$f(x) =\left ( {\sin }^{ -1 }x+{ \text{cosec} }^{ -1 }x\right )$$ is

lf $$f:[-6,6]\rightarrow \mathbb{R}$$ is defined by $$f(x)=x^{2}-3$$ for $$x\in \mathbb{R}$$ then
$$(fofof)(-1)+(fofof)(0)+(fofof)(1)=$$

lf $$f$$ : $$R\rightarrow R$$ is defined by
$$f(x)=\left\{\begin{array}{l}x+4 & x<-4\\3x+2 & -4\leq x<4\\x-4 & x\geq 4\end{array}\right.$$
then the correct matching of list I to List II is. 

The domain of $$\mathrm{f}(\mathrm{x})=\log_{\mathrm{x}}(9-\mathrm{x}^{2})$$ is

The domain of $$\displaystyle f(x)=\sqrt{\log\left(\frac{1}{|\sin x|}\right)}$$ is

lf $$g(f(x)) =|\sin \mathrm{x}|,f(g(x)) =(\sin\sqrt{\mathrm{x}})^{2}$$, then