Let \[f_n(x)=(2+(-2)^n)x^2+(n+3)x+n^2\] where \(n\) is a positive integer and \(x\) is any real number.

  1. Write down \(f_3(x)\).

    Find the maximum value of \(f_3(x)\).

    For what values of \(n\) does \(f_n(x)\) have a maximum value (as \(x\) varies)? [Note you are not being asked to calculate the value of this maximum].

  2. Write down \(f_1(x)\).

    Calculate \(f_1(f_1(x))\) and \(f_1(f_1(f_1(x)))\).

    Find an expression, simplified as much as possible, for \[f_1(f_1(f_1(...f_1(x))))\] where \(f_1\) is applied \(k\) times. [Here \(k\) is a positive integer.]

  3. Write down \(f_2(x)\).

    The function \[f_2(f_2(f_2(...f_2(x)))),\] where \(f_2\) is applied \(k\) times, is a polynomial in \(x\). What is the degree of this polynomial?