Consider a function f(x) which satisfies the followingproperties:
1. f(x+y)=f(x) * f(y)
2. f(0) does not equal to 0
3. f'(0)=1
Then:
a) Show that f(0)=1. (Hint: use the fact that 0+0=0)
b) Show that f(x) does not equal to 0 for all x. (Hint: use y=-x with conditions (1) and (2) above.)
c) Use the definition of the derivative to show that f'(x)=f(x)for all real numbers x
d) let g(x) satisfy properties (1)-(3) above and let k(x)=f(x)/g(x). Show that k(x) is defined for all x and find k'(x). Usethis to discover the relationship between f(x) and g(x)
e) Can you think of a function that satisfies (1)-(3)? Couldthere be more than one such function? Explain why or why not.
f) What do you think would happen if you changed condition (3)so that f'(0)=a for some a>0, rather than 1? Would you stillfind a familiar function?
