First slide

Functions (XII)

Question

Statement I : Every function must be either even or odd function
Statement II : The function $f (x) = \log (x + \sqrt{x^{2} + 1})$ is an odd function

Which of the following is true

Moderate

A

ony I
B

only II
C

both I and II
D

neither I nor II

Solution

I : is not true
Ex : $f (x) = x^{2} + x + 1$ then $f (- x) = x^{2} - x + 1 \neq f (x)$ or $\neq - f (x)$

II : $f (- x) = \log ((\sqrt{x^{2} + 1} - x) \frac{\sqrt{x^{2} + x} + x}{\sqrt{x^{2} + 1} + x})$

$= - \log (x + \sqrt{x^{2} + 1}) = - f (x)$

odd function

only II is true

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