ZA-SAMO 1996 Problem 6

The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$ . If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$ . The following are corrent to the number of digits given:

$$3^{0.25}=1.31607, 3^{0.50}=1.73205, 3^{0.75}=2.27951.$$