A function f, defined for all real X, has the following properties I) f(x)>0 for all X, ii)f(x+y)=f(x)f(y) for all X and y and iii) f(x)=f(y) if and only if x=y. Prove that a) f(0)=1, b) f(-x) =1/f(x) for all X c) f(x-y)=f(x)/f(y) for all X and y

Question

A function f, defined for all real X, has the following properties I) f(x)>0 for all X, ii)f(x+y)=f(x)f(y) for all X and y and iii) f(x)=f(y) if and only if x=y. Prove that a) f(0)=1, b) f(-x) =1/f(x) for all X c) f(x-y)=f(x)/f(y) for all X and y

Answer 1

a) To find f(0), notice that from property ii), we have f(0 + 0) = f(0)f(0). Since f(0) > 0 (property i), we can divide both sides of the equation by f(0) to get f(0) = 1.

b) To find f(-x), notice that from property ii) again, we have f(x + -x) = f(x)f(-x). Since f(x) > 0 and f(-x) > 0 (property i), we can divide both sides of the equation by f(x)f(-x) to get 1 = f(-x), and therefore f(-x) = 1/f(x) for all x.

c) To find f(x - y), notice that from property ii), we have f(x + (-y)) = f(x)f(-y). Now we can find f(-y) as we did in part b): f(-y) = 1/f(y). Plug this result back in our equation and get f(x - y) = f(x)/f(y) for all x and y.