by Samar Imam Zaidi
Last Updated September 21, 2018 04:20 AM

If $f(x)=\frac{x}{lnx}$ & $g(x)=\frac{lnx}{x}$. 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) $f(x).g(x)=1 \forall x>0$

D) $\frac{1}{f(x).g(x)}=1 \forall x>0$

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.

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