The function f(x) = ( (x + x^2 + 1 ) ) is - Vidyadip

MCQ

The function $f(x) = \sin \left( {\log (x + \sqrt {{x^2} + 1} )} \right)$ is

A
Even function
✓
Odd function
C
Neither even nor odd
D
Periodic function

✓

Answer

Correct option: B.

Odd function

b
(b) $f(x) = \sin \,\left( {\log \,(x + \sqrt {1 + {x^2}} )} \right)$

==> $f( - x) = \sin \,[\log \,( - x + \sqrt {1 + {x^2}} )]$

==> $f( - x) = \sin \,\log \left( {(\sqrt {1 + {x^2}} - x)\frac{{(\sqrt {1 + {x^2}} + x)}}{{(\sqrt {1 + {x^2}} + x)}}} \right)$

==> $f( - x) = \sin \,\log \,\left[ {\frac{1}{{(x + \sqrt {1 + {x^2}} )}}} \right]$

==> $f( - x) = \sin \left[ {\log {{(x + \sqrt {1 + {x^2}} )}^{ - 1}}} \right]$

==> $f( - x) = \sin \left[ { - \log (x + \sqrt {1 + {x^2}} )} \right]$

==> $f( - x) = - \sin \left[ {\log (x + \sqrt {1 + {x^2}} )} \right]$==> $f( - x) = - f(x)$

$f(x)$ is odd function.

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