Algebra

Functions

Functions: Level 5 Challenges

         

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

Then find the product of roots of the equation below.

[f(x)]2+3x f(x)+2x2=0 \large [f(x)]^{2} + 3x \ f(x) + 2x^{2} = 0

Let ff be a function from the integers to the real numbers such that f(x)=f(x1)f(x+1). f(x) = f(x-1) \cdot f(x+1).

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

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

f(x)f(2x2)=f(2x3+x)\large f(x)f(2x^2) = f(2x^3+x)

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

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

The functions f(x)f(x) and g(x)g(x) are defined R+R\mathbb {R^+ \to R} such that f(x)={1xx is rationalx2x is irrationalg(x)={x   x is rational1xx is irrationalf(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 fg(x)f \circ g(x) is

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