Answer
The point $~~1-1~i~~$ is not in the Mandelbrot set.
Work Step by Step
$z = 1-1~i$
We can perform the calculation $z^2+z$:
$z^2+z = (1-1~i)^2+(1-1~i)$
$z^2+z = (0-2~i)+(1-1~i)$
$z^2+z = 1-3~i$
We can perform the calculation $(z^2+z)^2+z$:
$z^2+z = (1-3~i)^2+(1-1~i)$
$z^2+z = (-8-6~i)+(1-1~i)$
$z^2+z = -7-7~i$
The absolute value is $\sqrt{(-7)^2+(-7)^2} = \sqrt{98}$ which is greater than 2.
Therefore, the point $~~1-1~i~~$ is not in the Mandelbrot set.
