Algebra

# Functional Equations - Introduction

Define the function $$f^1(x) = \frac{1}{1-x}$$, and $$f^n (x) = f^1 ( f^{n-1} (x) )$$ for positive integers $$n$$.

Evaluate $$f^{36} ( 10 ) .$$

Find all functions $$f: \mathbb{R} \rightarrow \mathbb{R}$$ such that

$$5 f( x + y) + y^ 5 = f(x) + (x+y) ^ 5.$$

Suppose that function $$f$$ satisfies

$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \text{ and } f(0)=17.$

Which of the following represents the family of solutions for $$f(x)$$?

$$k$$ in the choices below is a constant.

$$P(x)$$ is a monic polynomial of degree $$2017$$ If $P(0)=2016, P(1)=2015, P(2)=2014, \ldots, P(2016)=0,$ what is the value of $$P(2017)?$$

If function $$f$$ satisfies $$f(2)=12$$ and $f(x+y)=f(x)+f(y)$ for all real numbers $$x$$ and $$y,$$ what is the value of $$f(-3)?$$

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