Identical Function Query If  $f(x)=\frac{x}{\ln x}$  &   $g(x)=\frac{\ln x}{x}$. Then identify the correct statement.
A) $\frac{1}{g(x)}$ and $f(x)$ are identical functions
B) $\frac{1}{f(x)}$ and $g(x)$ are identical functions
C) $f(x)\cdot g(x)=1 \forall x>0$
D) $\frac{1}{f(x)\cdot g(x)}=1 \forall x>0$
I don't have the solution but as per the answer key Only A is the correct statement , B,C,D are incorrect statement .
My Approach for B let $t(x)=\frac{1}{f(x)}$ , now the question is whether $t(x)$ & $g(x)$ are identical function, my thought would be that they are identical function because for identical function we need to check domain and range on $t(x)$ and not on its reciprocal.But on contrary in the ANSWER Key this is mentioned as INCORRECT. 
Regarding C and D I don't know why it is incorrect.

The important point is that $\ln 1=0$, so you can't divide by it.  In A you get a $\ln x$ in both denominators, so both sides of the equation are undefined at $x=1$.  In B, $g(x)$ is nicely defined for all $x \gt 0$, but $f(1)$ is not so $\frac 1{f(1)}$ is not either.  C and D both fail for $x=1$ for the same reason.
