Fredrick wrote:

> ...what do we do when \\( x \lt 0 \\)?

We have a formula for the right adjoint if it exists: we saw it near the end of [Lecture 6]( So, we can use this to figure out what \\(R(x)\\) must be if the right adjoint exists... and if the formula gives an undefined result, we know the right adjoint cannot exist.

Another approach is to use Proposition 1.81 in _[Seven Sketches]( Applied to our example, this implies that if \\(R : \mathbb{R} \to \mathbb{N}\\) is a right adjoint to \\(I : \mathbb{N} \to \mathbb{R} \\), we must have

$$ I(R(x)) \le x $$

for all \\(x \in \mathbb{R}\\). See what this means?