Functional Equations

Functional Equations - Introduction


Define the function f1(x)=11x f^1(x) = \frac{1}{1-x} , and fn(x)=f1(fn1(x)) f^n (x) = f^1 ( f^{n-1} (x) ) for positive integers nn .

Evaluate f36(10). f^{36} ( 10 ) .

Find all functions f:RR f: \mathbb{R} \rightarrow \mathbb{R} such that

5f(x+y)+y5=f(x)+(x+y)5. 5 f( x + y) + y^ 5 = f(x) + (x+y) ^ 5.

Suppose that function ff satisfies

f(x+y2)=f(x)+f(y)2 and f(0)=17.f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} \text{ and } f(0)=17.

Which of the following represents the family of solutions for f(x) f(x) ?

kk in the choices below is a constant.

P(x)P(x) is a monic polynomial of degree 2017 2017 If P(0)=2016,P(1)=2015,P(2)=2014,,P(2016)=0,P(0)=2016, P(1)=2015, P(2)=2014, \ldots, P(2016)=0, what is the value of P(2017)?P(2017)?

If function ff satisfies f(2)=12f(2)=12 and f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for all real numbers xx and y,y, what is the value of f(3)?f(-3)?


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