MathDB

Find all monotonically increasing or monotonically decreasing functions

f: \mathbb{R}_+\to\mathbb{R}_+

which satisfy the equation

f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1

for any two numbers

x

and

y

from

\mathbb{R}_+

. Hereby,

\mathbb{R}_+

is the set of all positive real numbers. Note. A function

f: \mathbb{R}_+\to\mathbb{R}_+

is called monotonically increasing if for any two positive numbers

x

and

y

such that

x\geq y

, we have

f\left(x\right)\geq f\left(y\right)

. A function

f: \mathbb{R}_+\to\mathbb{R}_+

is called monotonically decreasing if for any two positive numbers

x

and

y

such that

x\geq y

, we have

f\left(x\right)\leq f\left(y\right)

Problem Statement