# Identical Function Query

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.

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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.