1975 AMC 12 Question 21

Suppose $f(x)$ is defined for all real numbers $x;$ $f(x)\gt0$ 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. $f(-a)=\frac{1}{f(a)}$ for all $a$

III. $f(a)=\sqrt[3]{f(3a)}$ for all $a$

IV. $f(b)\gt f(a)$ if $b\gt a$