Let \(f\) and \(g\) be two functions defined by \(f(x) = |x + |x||\) and \(g(x) = \frac{1}{x}\) for \(x \neq 0\). If \(f(a) + g(f(a)) = \frac{13}{6}\) for some real \(a\), then the maximum possible…

The maximum is 6. We need $a>0$ so that $f(a)=2a$ is nonzero, and with $t=2a$ the equation $t+1/t=13/6$ gives $t=2/3$ or $t=3/2$. Then $f(g(a))=2/a=4/t$, whose maximum comes from $t=2/3$. The value 3 is the reciprocal of the smaller $a$ before applying the outer function.