Answer:
[tex]f(\frac{2}{7}})=\frac{3}{8}[/tex]
Step-by-step explanation:
By properties i) and iii), [tex]f(1-0)=1-f(0)=1=f(1)[/tex]. Now we can use properties iii) iv) to compute some values of f. Namely:
[tex]f(\frac{1}{2})=f(1-\frac{1}{2})=1-f(\frac{1}{2}) \rightarrow f(\frac{1}{2})=\frac{1}{2} [/tex]
[tex]f(\frac{1}{3})=\frac{f(1)}{2}=\frac{1}{2}[/tex]
[tex]f(\frac{1}{6})=f(\frac{1}{2} \frac{1}{3})=\frac{f(\frac{1}{2})}{2}=\frac{1}{2}\frac{1}{2}=\frac{1}{4}[/tex]
[tex]f(\frac{1}{9})=f(\frac{1}{3} \frac{1}{3})=\frac{f(\frac{1}{3})}{2}=\frac{1}{2}\frac{1}{2}=\frac{1}{4}[/tex]
With these values, we can obtain f(1/7) using property ii). Note that:
[tex]\frac{1}{6}>\frac{1}{7}>\frac{1}{9} \rightarrow f(\frac{1}{6})\geq f(\frac{1}{7})\geq f(\frac{1}{9}) \rightarrow \frac{1}{4} \geq f(\frac{1}{7}) \geq \frac{1}{4}[/tex] then [tex]f(\frac{1}{7}})=\frac{1}{4}[/tex].
Finally, combine the previous work with properties iii) and iv) to get
[tex]f(\frac{2}{7})=f(\frac{6}{7} \frac{1}{3})=\frac{1}{2}f(\frac{6}{7})=\frac{1}{2} f(1-\frac{1}{7})=\frac{1}{2}(1-f(\frac{1}{7}))=\frac{1}{2} (1-\frac{1}{4})=\frac{3}{8}[/tex]
