Thread: Exponential help

Hi, I really need help on this problem.

Let f(x) = (1/2)([a^x] + [a^-x]) if a> 0. Show that

f(x + y) + f(x-y) = 2f(x)f(y)

I really appreciated any help

Originally Posted by BrianW

Hi, I really need help on this problem.

Let f(x) = (1/2)([a^x] + [a^-x]) if a> 0. Show that

f(x + y) + f(x-y) = 2f(x)f(y)

I really appreciated any help

f(x+y) = (1/2) (a^{x+y} + a^{-(x+y)})

f(x+y) = (1/2) (a^{x-y} + a^{-(x-y)})

so:

f(x+y) + f(x-y) = (1/2) [a^{x+y} + a^{-(x+y)} + a^{x-y} + a^{-(x-y)}]

........ = (1/2) [ a^x (a^y-a^{-y}) + a^{-x} (a^y-a^{-y})]

........ = f(y) [ a^x + a^{-x}] = 2 f(y) f(x) = 2 f(x) f(y)

RonL