Question

Consider a function f(x) which satisfies the following properties:

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). Use this to discover the relationship between f(x) and g(x)

e) Can you think of a function that satisfies (1)-(3)? Could there 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 still find a familiar function?

Answer 1