#1**+1 **

\(\text{a little toying with it shows that }\\ f^{\circ k}(x) = c,~\forall k >0\\ \text{here }f^{\circ k} \text{ means applying the function }f, ~k \text{ times, i.e.}\\ f^{\circ 2}(x) = f(f(x))\\ \text{I leave it to you to determine }c \text{ and convince yourself that above is true}\)

.Rom Feb 20, 2019

#2**+3 **

f(x)=x^2-2x How would you find f(f(f(f(f(f(-1)))))) ?

We (seemingly) have six evaluations to make

(1) f ( - 1) = (-1)^2 -2(-1) = 1 + 2 = 3

(2) f ( f (-1) ) = f (3) = (3)^2 - 2(3) = 9 - 6 = 3

(3) f ( f ( f (-1) ) ) = f(3) = 3

Note that this pattern continues....so

f(f(f(f(f(f(-1)))))) = 3

CPhill Feb 20, 2019