How does squaring give you a monotonic transformation?

Question

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?

Answer 1

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.

Answer 2

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 $[0,+\infty)$.