Functions Test ...

Consider the following statements :
Statement 1 : The function $$f:R \rightarrow R$$ such that $$f(x)=x^3$$ for all $$x\in R$$ is one-one.
Statement 2 : $$f(a) = f(b) \Rightarrow a=b$$ for all $$a, b \in R$$ if the function $$f$$ is one-one.
Which one of the following is correct in respect of the above statements?

If $$f(x) = 8x^3, g(x) = x^{1/3}$$, then fog (x) is

If $$fog = |\sin x|$$ and $$gof = \sin^{2}\sqrt {x}$$, then $$f(x)$$ and $$g(x)$$ are

If $$g(x)=\dfrac{1}{f(x)}$$ and $$f(x)=x, x\ne 0,$$ then which one of the following is correct?

Let $$f (x) = \sqrt {2 - x - x^2}$$ and g(x) = cos x. Which of the following statements are true?
(I) Domain of $$f((g(x))^2) = $$ Domain of f(g(x))
(II) Domain of f(g(x)) + g(f(x)) = Domain of g(f(x))
(III) Domain of f(g(x)) = Domain of g(f(x))
(IV) Domain of $$g((f(x))^3) = $$ Domain of f(g(x))

If $$g(x)=1+\sqrt{x}$$ and $$f\{g(x)\}=3+2\sqrt{x}+x$$, then $$f(x)$$ is equal to

If $$f:R\rightarrow R, g:R \rightarrow R$$ be two functions given by $$f(x)=2x-3$$ and $$g(x)=x^3+5$$, then $$(fog)^{-1}(x)$$ is equal to

The function $$f:A\rightarrow B$$ given by $$f(x) = x ,x\in A$$, is one to one but not onto. Then;

If $$f(x)=\sin ^{ 2 }{ x } +\sin ^{ 2 }{ \left( x+\cfrac { \pi  }{ 3 }  \right)  } +\cos { x } \cos { \left( x+\cfrac { \pi  }{ 3 }  \right)  } $$ and $$g\left( \cfrac { 5 }{ 4 }  \right) =1$$, then $$g\circ f(x)$$ is equal to

If $$f(x)=ax+b $$ and $$g(x)=cx+d$$, then $$f\left( g(x) \right) =g\left( f(x) \right) \Leftrightarrow$$