Functions
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.
\[ \large [f(x)]^{2} + 3x \ f(x) + 2x^{2} = 0 \]
by
Kunal Verma
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 \(f(x)\)?
by
Josh Speckman
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 \(f(5)\).
by
Parth Lohomi
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 \(f(0),f(1),f(2),\ldots,f(999)?\)
by
siddharth bhatt
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
by
Pranjal Jain