MCQ
Given the function $f (x)=\frac{ a + a }{2}, a > 2$, then
- ✓$2 f (x) \cdot f (y)$
- B$f (x) \cdot f (y)$
- C$\frac{ f (x)}{ f (y)}$
- D$\frac{1}{2} f (x) f (y)$
✓
Answer
Correct option: A.
$2 f (x) \cdot f (y)$
(A)
$f (x+y)+ f (x-y)$
$=\frac{1}{2}\left[ a ^{x+y}+ a ^{-x-y}+ a ^{x-y}+ a ^{-x+y}\right]$
$=\frac{1}{2}\left[ a ^x\left( a ^y+ a ^{-y}\right)+ a ^{-x}\left( a ^y+ a ^{-y}\right)\right]$
$=\frac{1}{2}\left( a ^x+ a ^{-x}\right)\left( a ^y+ a ^{-y}\right)$
$=2 f (x) f (y)$
$f (x+y)+ f (x-y)$
$=\frac{1}{2}\left[ a ^{x+y}+ a ^{-x-y}+ a ^{x-y}+ a ^{-x+y}\right]$
$=\frac{1}{2}\left[ a ^x\left( a ^y+ a ^{-y}\right)+ a ^{-x}\left( a ^y+ a ^{-y}\right)\right]$
$=\frac{1}{2}\left( a ^x+ a ^{-x}\right)\left( a ^y+ a ^{-y}\right)$
$=2 f (x) f (y)$
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