First slide

Functions (XII)

Question

which pair of functions is identical?

Moderate

A

$\sin^{- 1} (\sin x) and \sin (\sin^{- 1} x)$
B

$\log_{e} e^{x}, e^{\log_{e} x}$
C

$\log_{e} x^{2}, 2 \log_{e} x$
D

$None of these$

Solution

Here,
$(a) \sin^{- 1} (\sin x) is defined for x \in [- \frac{π}{2}, \frac{π}{2}], while \sin (\sin^{- 1} x) is defined only for x \in [- 1, 1]$
$(b) \log_{e} e^{x}, is defined for all x, while e^{\log_{e} x} is defined for x > 0 .$
$(c) \log_{e} x^{2} is defined for all x \in R - {0}, while 2 \log_{e} x is definedforx>0.$
Thus, none is identical.

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