Question

Which of the following functions are identical?

• A. $f\left(x\right)=\ln x^2\mathrm{and}g\left(x\right)=2\ln x$
• B. $f\left(x\right)={\log }_{x}e\mathrm{and}g\left(x\right)=\frac{1}{{\log }_{e}x}$
• C. $f\left(x\right)=\sin \left({\cos }^{-1}x\right)\mathrm{and}g\left(x\right)=\cos \left({\sin }^{-1}x\right)$
• D. None of these

Answer 1

Answer: (B), (C)

$f\left(x\right)=\sin \left({\cos }^{-1}x\right)\mathrm{and}g\left(x\right)=\cos \left({\sin }^{-1}x\right)$

$\mathrm{For}\mathrm{the}\mathrm{two}\mathrm{functions}\mathrm{to}\mathrm{be}\mathrm{identical},\mathrm{their}\mathrm{range}\mathrm{and}\mathrm{domain}\mathrm{should}\mathrm{be}\mathrm{same}.\to f\left(x\right)=\ln x^2\&g\left(x\right)=2\ln x\mathrm{the}\mathrm{domain}\mathrm{of}f\mathrm{is}R-\left(0\right\},\mathrm{while}\mathrm{the}\mathrm{domain}\mathrm{of}g\mathrm{is}{R}^{+}\mathrm{so},f\mathrm{and}g\mathrm{are}\mathrm{not}\mathrm{identical}\to f\left(x\right)={\log }_{x}e\&g\left(x\right)=\frac{1}{{\log }_{e}x},\mathrm{the}\mathrm{domain}\mathrm{of}f\mathrm{and}g\mathrm{is}{R}^{+}-\left(1\right\}\mathrm{and}f\left(x\right)={\log }_{x}e=\frac{1}{{\log }_{e}x}=g\left(x\right)\mathrm{so},f\mathrm{and}g\mathrm{are}\mathrm{identical}\to f\left(x\right)=\sin \left({\cos }^{-1}x\right)\&g\left(x\right)=\cos \left({\sin }^{-1}x\right),\mathrm{the}\mathrm{domain}\mathrm{of}f\mathrm{and}g\mathrm{are}[-1,1]\mathrm{and}f\left(x\right)=\sin \left({\cos }^{-1}x\right)=\sin (\frac{\pi }{2}-{\sin }^{-1}x)({\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2},\mathrm{for}x\in [-1,1\left]\right)=\cos \left({\sin }^{-1}x\right)=g\left(x\right)\mathrm{so},f\mathrm{and}g\mathrm{are}\mathrm{identical}$