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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."

Level 3

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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