Question

If $f(x)=\frac{x}{\ln x}$ and $g(x)=\frac{\ln x}{x}$. 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. $\frac{1}{f(x)\cdot g(x)}=1\forall x>0$

Answer 1

Answer: (A)

(A) $\frac{1}{g(x)}=\frac{1}{\ln n/x};f(x)=\frac{x}{\ln x}x>0,x\ne 1$ for both
(B) $\frac{1}{f(x)}=\frac{1}{x/\ln x};g(x)=\frac{\ln x}{x}\Rightarrow \frac{1}{f(x)}$ is not defined at $x=1$ but $g(1)=0$
(C) $f(x)\cdot g(x)=\frac{x}{\ln x}\cdot \frac{\ln x}{x}=1$ if $x>0,x\ne 1\Rightarrow$ N. I.
(D) $\frac{1}{f(x)\cdot g(x)}=\frac{1}{\frac{x}{\ln x}\cdot \frac{\ln x}{x}}=1$ only for $x>0$ and $x\ne 1]$