Functional Equations

Functional Equations - Substitution


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

f(aa1)11f(a1a)=0, f \left( \frac{a}{a-1} \right) - 11 f \left( \frac{ a-1} {a} \right) = 0,

for a0,1 a \neq 0, 1 .

A function f(x)f(x) from the positive integers to the reals satisfies
1) f(1)=792 f(1) = 792
2) f(1)+f(2)++f(n)=n2f(n) f(1) + f(2) + \ldots + f(n) = n^2 f(n) for any positive integer nn.

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

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

(x2x+1)f(x2)=f(x),f(3)=3.\left(x^2-x+1\right)f\left(x^2\right)=f(x), f(3)=3.

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

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

f(x)+f(11x)=5x, f(x) + f \left( \frac{1}{1-x} \right) = 5 x,

for x0,1 x \neq 0, 1 . What is the value of 40f(5) 40 f( 5)?

Suppose that ff satisfies the following functional equation: 2f(x)+3f(3x+29x3)=100x+80.2f(x)+3f\left(\frac{3x+29}{x-3}\right)=100x+80. What is the value of f(4)?f(4)?


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