Relations and F...

Let $$A=\left\{ x\in R|x\ge 0 \right\} $$. A function $$f:A\rightarrow A$$ is defined by $$f(x)={ x }^{ 2 }$$. Which one of the following is correct?

Let $$f(x)=\dfrac{x+1}{x-1}$$ for all $$x \neq 1$$. 

If $$f : (3, 4) \rightarrow (0, 1)$$ is defined by $$f(x)=x-\left[x\right]$$, where $$\left[x\right]$$ denotes the greatest integer function, then $${f}^{-1}(x)$$ is

Let $$M$$ be the set of all $$2 \times 2$$ matrices with entries from the set of real numbers $$R$$. Then the function $$ f : M \rightarrow R$$ defined by $$f\left( A \right) =\left| A \right|$$ for every $$ A\in M$$, is

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?

On the set $$Z$$, of all integers $$\ast$$ is defined by $$a\ast b = a + b - 5$$. If $$2\ast (x\ast 3) = 5$$ then $$x =$$

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

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

If $$f : R \rightarrow R$$ is defined by $$f(x) = x^{3}$$ then $$f^{-1}(8) =$$