×
Back to all chapters

# Functions

Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.

# 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

×