0<x<g (x) = x+1(4x) ∈ [1,+∞) When = 1/2,

1/2 & lt； X<G (x) ∈ (1, 5/4) is at 1,

X & ltG(x)=x+ 1∈(-∞, 1) When =0.

The equation g[f(x)]-a=0 has four different real roots:

When a< is 1, f (x) = a- 1 =-x 2-2x has only two real roots;

When a= 1, f(x)=0 or 1/2, with four real roots;

1 & lt； A< in 5/4, f(x)=[a soil √ (A 2- 1)]/2 =-X 2-2x, with four real roots;

A> when =5/4, f(x)=[a soil √ (A 2- 1)]/2 =-X 2-2x, with 2 or 3 real roots.

To sum up, 1