Let f x = a x ( a 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:

Question

Let f x = a x ( a > 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:

TestHub · Accepted Answer

f x = a x , a > 0 f x = a x + a - x + a x - a - x 2 ⇒ f 1 x = a x + a - x 2 f 2 x = a x - a - x 2 ⇒ f 1 x + y + f 1 x - y = a x + y + a - x - y 2 + a x - y + a - x + y 2 = a x + a - x 2 a y + a - y = f 1 x × 2 f 1 y = 2 f 1 x f 1 y

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