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Let a function is defined as f(x)=(a^(x)...

Let a function is defined as `f(x)=(a^(x)+a^(-x))/(2), a gt 0`, what is the value of `f(x+y)+f(x-y)`?

A

`2 f (x) . F(y)`

B

`f(x) .f(y)`

C

`f(x) //f(y)`

D

None

Text Solution

Verified by Experts

The correct Answer is:
A
Given `f(x) = (a^(x)+a^(-x))/(2)`.....(1)
`:. f(y) = (a^(y)+a^(-y))/(2)`
`f(x + y) = (a^(x+y)+a^(-(x+y)))/2 ; f(x − y) = (a^(x-y)+a^(-(x-y)))/(2)`
`f(x + y) + f(x − y) = (a^(x+y)+a^(-(x+y))+a^(x-y)+a^(-(x-y)))/(2)`
`= (a^xa^y+a^(-x)a^(-y)+a^x.a^(-y)+a^(-x).a^y)/2 =(a^y .(a^x+a^(-x))+a^(-y)(a^x+a^(-x)))/2`
`=2.(a^x+a^(-x))/2 . (a^y+a^(-y))/2`
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