Iteration basics

[Gottfried]

Common difficulty. Sometime brain blocks...

Assume

Code:

´
f(x) = x^2+2

Code:

´
a) f(x)^2 = (x^2+2)^2           // taking f to the 2'nd power
          =  x^4 + 4x^2 +4

b) f°2(x) = f(f(x))             // iteration of f
          = (f(x))^2  + 2
          =  (x^4 + 4x^2 +4) + 2
          = x^4+4x^2+6

c)  e^f(x) = ...                // exponentiation of f

Assume

Code:

´
       f(x) = e^x
            = 1 + x/1! + x^2/2! + ....

Code:

´
a)       f(x)^2 = (1 + x/1! + x^2/2! +...)^2   // taking f to the second power
                =  1 + 2*x/1! + 4*x^2/2! + 8*x^3/3! + 16*x^4/4! +...
                = f(2x)

b)       f°2(x) = f(f(x))                      // iteration of f
                = f(e^x)
                = 1 + e^x/1! + e^2x/2! + ...

c)       e^f(x) = ...                          // exponentiation of f
                     // is in this case equal to second iteration

(You had this right previous times )

Gottfried