Functional Equations

Functional Equations: Level 3 Challenges


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

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

f(f(x))=6x+f(x). f(f(x) ) = 6x + f(x).

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

A function f:RRf: \mathbb{R} \rightarrow \mathbb{R} satisfies f(5x)=f(5+x) f (5-x) = f(5+x) . If f(x)=0f(x) = 0 has 55 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 x^* such that f(x)=0 f(x^*) =0.

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

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

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

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

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


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