Let be a function piecewise defined as If is negative, find so that ​ .

 a = -30.5:

g(10.5) = -20; g(-20) = 20; g(20) = -1 

a = -30.5:  g(-30.5) = 30.5;  g(30.5) = 20;  g(20) = -1

Edited to correct silly mistake!  Thanks for pointing this out Melody.

Thanks Alan 

I find these difficult to get my head around too.

I find it easier if I draw a graph and I can see what I am doing visually.

g(10.5)=-20

g(-20)=20

g(20)=-1

          Now going backwards.

         g(what)=20    there are two answers, -20 and 30.5   (this is easy to see from the graph)

          so

          g(-20)=20      and     g(30.5)=20

                   now go backwards again

                   g(what)=-20      and     g(what)=30.5  looking at the graph I can see that

                   g(10.5)=-20      and      g(-30.5)=30.5    and   g(35.75)=30.5

so

g(10.5) is what we started with

You are told that a is negative so  35.75 is no good

a=-30.5

check

g(-30.5)=30.5

g(30.5)=20

g(20)=-1  = g(g(g(10.5)))

ie

g(g(g(-30.5)=g(g(g(10.5)))

a=-30.5