Quasiconcavification
In one dimension if the functions $f_i$ are bounded, e.g. defined on a compact interval, then yes. Take $g_1$ such that it makes $f_1$ almost constant, e.g. $g_1(x)=\frac{x}{N(\max f_1-\min f_1)}$ where $N&gt;0$ is large. Take $g_2(x)=x$ Then $g_1\circ f_1+g_2\circ f_2\approx f_2$, which is strictly quasiconcave by assumption. 

In one dimension a strictly quasiconcave function is either strictly increasing, strictly decreasing or single-peaked.