MathDB

Real function

f

generates real function

g

if there exists a natural

k

such that

f^k=g

and we show this by

f \rightarrow g

. In this question we are trying to find some properties for relation

\rightarrow

, for example it's trivial that if

f \rightarrow g

and

g \rightarrow h

then

f \rightarrow h

.(transitivity)
(a) Give an example of two real functions

f,g

such that

f\not = g

f\rightarrow g

and

g\rightarrow f

. (b) Prove that for each real function

f

there exists a finite number of real functions

g

such that

f \rightarrow g

and

g \rightarrow f

. (c) Does there exist a real function

g

such that no function generates it, except for

g

itself? (d) Does there exist a real function which generates both

x^3

and

x^5

? (e) Prove that if a function generates two polynomials of degree 1

P,Q

then there exists a polynomial

R

of degree 1 which generates

P

and

Q

.
Time allowed for this problem was 75 minutes.

Problem Statement