F:=x->x+x^2+lambda;
step1:=simplify((F(F(x))-x)/(x^2+lambda));
solns:=solve(step1,x);
Fp:=x->1+2*x;
simplify(Fp(solns[1])*Fp(solns[2]));
solve(-1<5+4*lambda and 5+4*lambda < 1,lambda);
F3:=collect(simplify(expand(F(F(F(x))))),x);
F3partial:=(simplify(((F(F(F(x))))-x)/((F(x))-x)));
simplify(subs(x=-1+sqrt(-1-lambda),F(F(F(F(x))))=x));
with(plots):
plot0:=implicitplot(((F(x)))=x, lambda=-3..0,x=-4..2,numpoints=10000,color=blue, thickness=4, color=BLACK):
plot1:=implicitplot((F(F(x)))=x, lambda=-3..0,x=-4..2,numpoints=10000,color=blue, thickness=3):
plot2:=implicitplot(F(F(F(F(x))))=x, lambda=-2.5..-3/2,x=-5..2,numpoints=50000,thickness=2):
plot3:=implicitplot(F(F(F(F(F(F(x))))))=x, lambda=-3..-3/2,x=-4..2,numpoints=50000,color=GREEN, thickness=1):
plot4:=implicitplot((((F(F(F(x))))))=x, lambda=-3..-3/2,x=-4..2,numpoints=50000,color=GREEN, thickness=1):
plot5:=implicitplot(F(F(F(F(F(F(F(x)))))))=x, lambda=-3..-3/2,x=-4..2,numpoints=50000,color=BLACK, thickness=1):
display({plot0,plot1,plot2,plot3});
display({plot0,plot1,plot2,plot4});
evalf(solve(subs({lambda=-1.99},F(F(F(x)))=x),x));
evalf(solve(subs({lambda=-2.0},F(F(F(x)))=x),x));
evalf(solve(subs({lambda=-2.1},F(F(F(x)))=x),x));
display({plot0,plot4,plot5});