Question

For which pair of functions is (g o f)(a)=|a|-2

Answer 1

There are infinitely many such pairs. Two simple ones:

1) f(a) = |a|, g(x) = x - 2. Then (g o f)(a) = g(f(a)) = f(a) - 2 = |a| - 2.

2) f(a) = a, g(x) = |x| - 2. Then (g o f)(a) = g(a) = |a| - 2.

(Also trivial: f(a) = |a| - 2 and g(x) = x gives g(f(a)) = |a| - 2.)