Given the function f(x) = a^x + a^ - x 2 , ;(a > 2). Then f(x + y… - Vidyadip

MCQ

Given the function $f(x) = \frac{{{a^x} + {a^{ - x}}}}{2},\;(a > 2)$. Then $f(x + y) + f(x - y) = $

✓
$2f(x).f(y)$
B
$f(x).f(y)$
C
$\frac{{f(x)}}{{f(y)}}$
D
None of these

✓

Answer

Correct option: A.

$2f(x).f(y)$

a
(a) We have $f(x + y) + f(x - y)$

$ = \frac{1}{2}\,\left[ {{a^{x + y}} + {a^{ - x - y}} + {a^{x - y}} + {a^{ - x + y}}} \right]$

$ = \frac{1}{2}\,\left[ {{a^x}({a^y} + {a^{ - y}}) + {a^{ - x}}({a^y} + {a^{ - y}})} \right]$

$ = \frac{1}{2}\,({a^x} + {a^{ - x}})\,\,({a^y} + {a^{ - y}}) = 2f(x)\,f(y)$.

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