Problem: Suppose f(x) is defined for all real numbers x;f(x)>0 for all x; and f(a)f(b)=f(a+b) for all a and b. Which of the following statements are true?
I. f(0)=1
II. f(−a)=1/f(a) for all a
III. f(a)=3f(3a)​ for all a
IV. f(b)>f(a) if b>a
Answer Choices:
A. III and IV only
B. I, II and IV only
C. All are true.
D. I, III and IV only
E. I, II and III only
Solution:
Letting a=0 in the equation f(a)f(b)=f(a+b) (called a functional equation) yields f(0)f(b)=f(b), or f(0)=1; letting b=−a in the functional equation yields f(a)f(−a)=f(0), or f(−a)=1/f(a); and f(a)f(a)f(a)=f(a)f(2a)=f(3a), or f(a)=3f(3a)​. The function f(x)=2−x satisfies the functional equation, but does not satisfy condition IV.