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Algebra

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 \( f^{36} ( 10 ) . \)

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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