Piecewise function

f(1)=-1

\(y=-x^2\), and \(y=1\)

\(1=-x^2\), solve for x, \(x=-1\), check it fits piecewise, and it does.

therefore \(f(1)=-1\)

You can also graph the equations \(y=-x^2\) with \(x \geq0\)  and \(y=x+8\) with \(x<0\), go to where \(y=1\) and see what \(x\) is.

f(1)  =   - (1)^2  =  -1

f (f(1) )  =  f (-1)  = -1 + 8  =  7

f (f (f (1) ) )  =  f ( 7)  = - (7)^2  = -49

 f ( f ( f( f(1) ) ) )   =  f (-49)  = -49 + 8  = -41

f ( f ( f ( f ( f ( 1) ) ) ) )  =  f (-41)  =  - (-41)^2  = -1681