Taking this example into account, we can see that setting the first value equal to 1, we obtain that
![F(x)=0.5x+1=4](https://tex.z-dn.net/?f=F%28x%29%3D0.5x%2B1%3D4)
and
![x=6](https://tex.z-dn.net/?f=x%3D6)
. Using this information, we find that
![F(x+1)=0.5(x+1)+1=0.5(6+1)+1=4.5](https://tex.z-dn.net/?f=F%28x%2B1%29%3D0.5%28x%2B1%29%2B1%3D0.5%286%2B1%29%2B1%3D4.5)
. It shows that when x is positive, the succussive terms are increasing.
Referring to that finding, if we set initial value less than zero, which means that we are solving 0.5x+1<0 and taking a number in the interval of the solution, which means x ∈ (- ∞, -20). Setting x=-19, we find that
![F(x)=0.5x+1=-19](https://tex.z-dn.net/?f=F%28x%29%3D0.5x%2B1%3D-19)
and x=-40. In the next iteration,
![F(x+1)=0.5(x+1)+1=0.5(1-40)+1=-18.5](https://tex.z-dn.net/?f=F%28x%2B1%29%3D0.5%28x%2B1%29%2B1%3D0.5%281-40%29%2B1%3D-18.5)
. In the next iteration,
![F(x+2)=0.5(x+2)+1=0.5(2-40)+1=-18](https://tex.z-dn.net/?f=F%28x%2B2%29%3D0.5%28x%2B2%29%2B1%3D0.5%282-40%29%2B1%3D-18)
. By this way, we find that even if the initial value is less than zero, value of the successive iterations is increasing.
Using the function
![g(x)=-x+2](https://tex.z-dn.net/?f=g%28x%29%3D-x%2B2)
and taking the initial value equal to 4, we find that
![g(x)=-x+2=4](https://tex.z-dn.net/?f=g%28x%29%3D-x%2B2%3D4)
and x=-2. In the next iteration,
![g(x+1)=-(x+1)+2=-(-2+1)+2=3](https://tex.z-dn.net/?f=g%28x%2B1%29%3D-%28x%2B1%29%2B2%3D-%28-2%2B1%29%2B2%3D3)
. If we continue the iterations we'll see that they are decreasing.
Setting the initial value equal to 2, we find that
![g(x)=-x+2=2](https://tex.z-dn.net/?f=g%28x%29%3D-x%2B2%3D2)
and x=0. The next iteration is
![g(x+1)=-(x+1)+2=1](https://tex.z-dn.net/?f=g%28x%2B1%29%3D-%28x%2B1%29%2B2%3D1)
. In this case, the interations are also decreasing.
If we set the initial value equal to 1, we find that
![g(x)=-x+2=1](https://tex.z-dn.net/?f=g%28x%29%3D-x%2B2%3D1)
and x=1. In the next iteration,
![g(x+1)=-(x+1)+2=0](https://tex.z-dn.net/?f=g%28x%2B1%29%3D-%28x%2B1%29%2B2%3D0)
and the iterations are decreasing.