Functional Equations

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 $f^{36} ( 10 ) .$

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

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

No solutions

f(x) =5 x^ 5

f(x) = x^ 5

f(x) = 0

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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.

ke^x+17-k

k\sqrt{x}+17

kx+17

kx^2+17

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$P(x)$ 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 $P(2017)?$

{2016}

{2017}!-1

-{2017}

{2017}!+{2016}

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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 $f(-3)?$

-24 -6 -12 -18

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