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 even function and ` f _ 2 (x) ` is an odd function. Then ` f _ 1 ( x + y ) + f _ 1 ( x - y ) ` equals :

Options

A. ` 2 f _ 1 (x ) f _ 2 (x ) `
B. ` 2 f _ 1 (x + y) f _ 2 ( x - y ) `
C. `2 f _ 1 ( x + 1 ) f _ 1 (x - y ) `
D. `2 f_ 1 ( x ) f _ 1 ( y) `

Explanation

To solve the problem, we need to express the function \( f(x) = a^x \) as a sum of an even function \( f_1(x) \) and an odd function \( f_2(x) \). Then, we will find the expression for \( f_1(x+y) + f_1(x-y) \).

### Step-by-step Solution:

1. **Identify the functions**:
We know that \( f(x) = a^x \). We can express \( f(x) \) as:
\[
f(x) = f_1(x) + f_2(x)
...