Functions

Algebra Level 3

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. If $f:{ \mathbb{Q} }_{ >0 } \rightarrow {\mathbb{Q} }_{ >0 }$ satisfies

$\large f\left( x^{ 2 }{ f\left( y \right) }^{ 2 } \right) = {f(x)}^2f\left( y \right) ,$

evaluate $f(2019).$

The answer is 1.

1 solution

Tristan Goodman
Aug 4, 2019

Solving the given equation algebraically results in f(x)=sqrt(x) but this does not satisfy the condition of all input giving a rational output, since for example sqrt(2) is irrational, so the function has to be either 1 or 0, but the conditions state that the function must have domain and range greater than zero; it must therefore be a constant function equal to 1.

Functions

The answer is 1.

1 solution

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