Review question

# Can we compose these polynomial functions? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7553

## Question

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?