Functions
Polynomial functional equation — composition and inverse integral
MJAT_TS3_P2
Grade 12
Question:
Let $f(x)$ be a polynomial satisfying the relation $f(f(f(x))) + (1-p)f(x) = 3,\ \forall x\in\mathbb{R}$, where $p$ is any real number. If the leading coefficient of $f(x)$ is $2$, then find the value of:
$$\left.\frac{d}{dx}\bigl(f(f(x))\bigr)\right|_{x=p} + \int_{-p}^{p}\bigl(2f^{-1}(x)+1\bigr)\,dx$$
Step-by-Step Solution
Key Concept: Since $\deg(f(f(f(x))))=\deg(f(x))$, we get $n^3=n$, so $n=1$. Thus $f(x)=2x+b$. From $f(f(f(x)))+(1-p)f(x)=3$: $8x+7b+(1-p)(2x+b)=3$. Matching: $8+2(1-p)=0\Rightarrow p=5$ and $b=1$.
$p=5$, $b=1$, $f(x)=2x+1$. Derivative $=4$, integral $=0$. Answer: $\mathbf{4}$.
Correct Answer: 4