Functions: Level 5 Challenges


Let \( f(x) \) be a cubic polynomial such that \( f(1) = 5, f(2) = 20, f(3) = 45 \).

Then find the product of roots of the equation below.

\[ \large [f(x)]^{2} + 3x \ f(x) + 2x^{2} = 0 \]

Let \(f\) be a function from the integers to the real numbers such that \[ f(x) = f(x-1) \cdot f(x+1). \]

What is the maximum number of distinct values of \(f(x)\)?

Let \(f(x)\) be a polynomial. It is known that for all \(x\),

\[\large f(x)f(2x^2) = f(2x^3+x)\]

If \(f(0)=1\) and \(f(2)+f(3)=125\), find \(f(5)\).

Given a function \(f\) for which \[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\] holds for all real \(x\), what is the largest number of different values that can appear in the list \(f(0),f(1),f(2),\ldots,f(999)?\)

The functions \(f(x)\) and \(g(x)\) are defined \(\mathbb {R^+ \to R}\) such that \[f(x)=\begin{cases} 1-\sqrt{x}\quad \text{x is rational} \\ \quad x^2\quad\quad\text{x is irrational}\end{cases}\\g(x)=\begin{cases} \quad x\quad\quad~~~ \text{x is rational} \\ 1-x\quad\quad\text{x is irrational}\end{cases}\]

The composite function \(f \circ g(x)\) is


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