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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."

# Solve by Substitution

Find all functions $$f: \mathbb{R} \backslash \{ 0, 1 \} \rightarrow \mathbb{R}$$ such that

$f \left( \frac{a}{a-1} \right) - 11 f \left( \frac{ a-1} {a} \right) = 0,$

for $$a \neq 0, 1$$.

A function $$f(x)$$ from the positive integers to the reals satisfies
1) $$f(1) = 792$$
2) $$f(1) + f(2) + \ldots + f(n) = n^2 f(n)$$ for any positive integer $$n$$.

What is the value of $$f(11)$$?

Consider the function $$f(x)$$ satisfying

$\left(x^2-x+1\right)f\left(x^2\right)=f(x), f(3)=3.$

If the value of $$f(-9)$$ can be expressed as $$\frac{a}{b},$$ where $$a$$ and $$b$$ are coprime positive integers, what is the value of $$a+7b?$$

A function $$f: \mathbb{R} \backslash \{ 0, 1 \} \rightarrow \mathbb{R}$$ satisfies

$f(x) + f \left( \frac{1}{1-x} \right) = 5 x,$

for $$x \neq 0, 1$$. What is the value of $$40 f( 5)$$?

Suppose that $$f$$ satisfies the following functional equation: $2f(x)+3f\left(\frac{3x+29}{x-3}\right)=100x+80.$ What is the value of $$f(4)?$$

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