Functional Equation and Composition

Given $f(x) + f(x-1) = 1$. Substituting $x-1$ into the relation: $f(x-1) + f(x-2) = 1$. Subtracting the two equations gives $f(x) - f(x-2) = 0 \implies f(x) = f(x-2)$. Thus $f$ is periodic with period 2. So $f(n)$ depends only on whether $n$ is even or odd. Note $x^2 - x = x(x-1)$ is always an even integer for any integer $x$. So $f(\text{even}) = 5$. Since $f(\text{even}) + f(\text{odd}) = 1$, $5 + f(\text{odd}) = 1 \implies f(\text{odd}) = -4$. Now $g(5) = 5^2 = 25$ (odd). So $f(g(5)) = f(25) = -4$. Also $f(5) = -4$, so $g(f(5)) = g(-4) = (-4)^2 = 16$. Therefore, $f(g(5)) + g(f(5)) = -4 + 16 = 12$.