Find the first three iterates of the function f(x)=x^2-2, if the initial value is -2?

Answer 1

2
2
2
x²-x-2=(x-2)(x+1) so if x=2 f(x)=x

Answer 2

What iterates are you referring to?
Do you mean Newton iterations?
If so, recall that Newton's formula gives
x_(n+1) = x_n -f(x_n)/f'(x_n)
Here f(x_n) = x_n²-2
f'(x_n) = 2x_n.
Here x_1 = -2
So x_2 = -2 - 2/-4 = -2 + 1/2 = -1.5
I'll let you carry on from here!