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Functional Equations

Functional Equations: Level 3 Challenges


If \(f\) is a function satisfying \(f(x+y) = 3^yf(x)+2^xf(y)\) for all \(x,y \in R\) and \(f(1) = 1\), what is the value of \(f(3)\)?

The polynomial function \( f: \mathbb{R}^+ \rightarrow \mathbb{R}^+\) satisfies the functional equation

\[ f(f(x) ) = 6x + f(x). \]

What is the value of \( f(17) \)?

A function \(f: \mathbb{R} \rightarrow \mathbb{R}\) satisfies \( f (5-x) = f(5+x) \). If \(f(x) = 0 \) has \(5\) distinct real roots, what is the sum of all of the distinct real roots?

Details and assumptions

The root of a function is a value \( x^*\) such that \( f(x^*) =0\).

Let a function \(f(x)\) has the property:

\[ f(x+2)= \frac{ f(x)-5}{f(x)-3}. \]

Then, which of the following must be equal to \( f(2014) \)?

Let \(f : \mathbb R^+ \rightarrow \mathbb R \) such that \[f(x)+2f \left (\frac{1331}{x} \right )=L \cdot x \] where \(L\) is a constant.

Knowing that \(f(11)=847\), what is \(L \)?


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