Relations and F...

Let R be a relation on the set of integers given by $$ aRb \leftrightarrow a = 2^k .b $$ for some integer $$k.$$ Then $$R$$ is 

Let $$g(x) = f(\log x) + f(2 - \log x)$$ and $$f''(x) < 0\forall x\in (0, 3)$$. Then find the interval in which $$g(x)$$ increases.

The domain of the function $$f(x) = \frac{x^2}{\sqrt{(a - x)(x-b)}}, (b > a)$$ is

Let R be a relation from $$ A=\{1,2,3,4\}$$ to $$B=\{1,3,5\}$$such that R=[(a,b):a<b where a\in{A} and b\in{B}].What is $$RoR^{-1} $$ equal to

COMPREHENSION :
If $${\left( {f\left( x \right)} \right)^2} \times f\left( {\frac{{1 - x}}{{1 + x}}} \right) = 6x,x \ne 0,1$$, then $$f(x)$$ is equal to 

Let $$f(-2, 2)\rightarrow(-2, 2)$$ be a continuous function given $$f(x)=f{(x}^{2})$$. Given $$f(0)=\dfrac{1}{2}$$ then the $$4f(\dfrac{1}{2})$$

If $$f:R \to R$$ satisfies $$f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$ for all $$x,y \in R$$ and $$f\left( 1 \right) = 7$$, then $$\sum\limits_{r = 1}^n {f\left( r \right)} $$

COMPREHENSION :
If $${\left( {f\left( x \right)} \right)^2} \times f\left( {\frac{{1 - x}}{{1 + x}}} \right) = 6x,x \ne 0,1$$, The domain of $$f(x)$$ is

If $$f(x)=1+x+\frac{x^2}{2} +...+\frac{x^{100}}{100}$$, then f'(1) is equal to

Let $$3f(x)-2f\left(\dfrac {1}{x}\right)=x$$, then $$f'(2)$$ is equal to