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Functional Equations

Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."

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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