First slide

Functions (XII)

Question

Which of the following functions are identical?

Moderate

A

$f (x) = \ln x^{2} and g (x) = 2 \ln x$
B

$f (x) = \log_{x} e and g (x) = \frac{1}{\log_{e} x}$
C

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

None of these

Solution

(1) For $f (x) = \log x^{2}, x^{2} > 0 or x \in R - {0}$
For $g (x) = 2 \log x, x > 0$
Hence, $f (x) and g (x)$ are not identical.
(2) $f (x) = \log_{x} e = \frac{1}{\log_{e} x} = g (x)$
Hence, the functions are identical.
(3) $f (x) = \sin (\cos^{- 1} x) = \sin (\frac{π}{2} - \sin^{- 1} x) = \cos (\sin^{- 1} x) = g (x)$
Hence, the functions are identical.

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