First slide

Functions (XII)

Question

A function whose graph is symmetrical about the origin is given by

Moderate

A

f (x) = (3^x + 3^{– x})
B

$f (x) = \cos [\log (x + \sqrt{1 + x^{2}})]$
C

f (x + y) = f (x) + f (y) ∀ x, y ∈ R
D

None of these

Solution

A function whose graph is symmetrical about the origin must be odd.
(3^x + 3^–x) is an even function.
Since cos x is an even function and $\log (x + \sqrt{1 + x^{2}})$ is an odd function,
$∴ \cos (\log (x + \sqrt{1 + x^{2}}))$ is an even function.
If f (x + y) = f (x) + f (y) ∀ x, y ∈ R, then f (x) must be odd.

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