\displaystyle
f_1(x)=1, f_2(x)=x, f_3(x)=x^2\\
f(f_1)=1, f(f_2)=\frac{3}{2}, f(f_3)=\frac{1}{3}\\
q(x)=a + b x + c x^2\\
\left< f_1| q \right> =2 a+\frac{2 c}{3}\\
\left< f_2| q\right> =\frac{2 b}{3}\\
\left< f_3| q \right> =\frac{2 a}{3}+\frac{2 c}{5}\\
a=1/2, b=9/4, c=0