A) \[{{\sin }^{-1}}(sinx)\,\,and\,\,sin\,(si{{n}^{-1}}x)\]
B) \[{{\log }_{e}}{{e}^{x}},{{e}^{{{\log }_{e}}x}}\]
C) \[{{\log }_{e}}{{x}^{2}},2lo{{g}_{e}}x\]
D) None of these
Correct Answer: D
Solution :
[d] Here,| (1) \[{{\sin }^{-1}}(\sin \,x)\] is defined for \[x\in \left[ -\frac{\pi }{2},\frac{\pi }{2} \right]\], while \[\sin (si{{n}^{-1}}x)\] is defined only for \[x\in [-1,1]\] |
| (2) \[{{\log }_{e}}{{e}^{x}},\]is defined for all x, while \[{{e}^{{{\log }_{e}}x}}\]is defined for \[x>0.\] |
| (3) \[{{\log }_{e}}{{x}^{2}}\]is defined for all \[x\in R-\{0\}\], while \[2{{\log }_{e}}x\]is defined for \[x>0.\] |
| Thus, none is identical. |
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