# How does squaring give you a monotonic transformation?

Consider the function: $$f(x,y) = \sqrt {xy}$$ Is the function $$f_1(x,y) = x^2 y^2$$ a monotonic transformation of $$f$$?

I remember studying earlier that squaring does not give you a monotonic transformation since the order will not be preserved for negative values of, say, $$x$$. But my textbook says that $$f_1$$ is a monotonic transformation.

I understand that we're actually squaring twice here -- but that still won't preserve the order for negative values, right?

## 2 Answers

There are no negative values. Under the usual interpretation of the $$\sqrt\$$ symbol, $$\sqrt {xy}$$ is zero or the positive square root of $$xy$$. And on $$[0,\infty)$$, the map $$t\longmapsto t^2$$ is monotone.

There are no negative values in the first place since the object you are squaring is the square root of something. The symbol $$\sqrt .$$ is always assumed to output a non-negative number.

Note that the map $$x\mapsto x^2$$ is monotonic if the domain is defined to be $$[0,+\infty)$$.