Solution
= Solution
Take
$$
f_1(n)=2n,\quad g_1(n)=n,\quad f_2(n)=g_2(n)=2^n.
$$
The premises hold with constants two and one. But
$$
\frac{f_2(f_1(n))}{g_2(g_1(n))}=\frac{2^{2n}}{2^n}=2^n\longrightarrow\infty.
$$
No fixed bounding constant can satisfy the proposed conclusion. This proves the <failure of unrestricted Big O composition>: a constant factor in an inner argument can become an unbounded factor after applying an exponential.