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Let f(x)=a^(x)(a gt 0) be written as f(x...

Let `f(x)=a^(x)(a gt 0)` be written as `f(x)=f_(1)(x)+f_(2)(x), " where " f_(1)(x)` is an function and `f_(2)(x)` is an odd function. Then `f_(1)(x+y)+f_(1)(x-y)` equals

A

`2f_(1)(x+y)*f_(2)(x-y)`

B

`2f_(1)(x+y)*f_(1)(x-y)`

C

`2f_(1)(x)*f_(2)(y)`

D

`2f_(1)(x)*f_(1)(y)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, function `f(x)=a^(x),a gt 0` is written as sum of an even and odd function `f_(1)(x) and f_(2)(x)` respectively.
Clearly, `f_(1)(x) =(a^(x)+a^(-x))/(2) and f_(2)(x)=(a^(x)-a^(-x))/(2)`
So, `f_(1)(x+y)+f_(1)(x-y)`
`=(1)/(2)[a^(x+y)+a^(-(x+y))]+(1)/(2)[a^(x-y)+a^(-(x-y))]`
`=(1)/(2)[a^(x)a^(y)+(1)/(a^(x)a^(y))+(a^(x))/(a^(y))+(a^(y))/(a^(x))]`
`=(1)/(2)[a^(x)(a^(y)+(1)/(a^(y)))+(1)/(a^(x))((1)/(a^(y))+a^(y))]`
` =(1)/(2)(a^(x)+(1)/(a^(x)))(a^(y)+(1)/(a^(y)))`
` =2((a^(x)+a^(-x))/(2))((a^(y)+a^(-y))/(2))= 2f_(1)(x)*f_(1)(y)`
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