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)` equalsA. `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)`

Let `f(x)=a^(x)(a gt 0)` be written as `f(x)=f_(1)(x)+f_(2)(x), " wher

1.	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)` equalsA. `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)`
Answer» Correct Answer - 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)`