Consider $$f(x) = \sqrt{x^2 - 5x + 4}$$
With nested function: $$x^2 - 5x + 4 = (x-1)(x-4)$$
$(x-1)(x-4)\,$ has to be positive in order for $f$ to be defined on the reals. So either both terms have to be positive or both terms have to be negative, or so I thought. Why do both have to be positive?
Is it this?
$$\sqrt{x^2 - 5x + 4} = \sqrt{(x-1)(x-4)} = \sqrt{(x-1)}\sqrt{(x-4)}$$ That they can be decomposed into single term square roots?
This can be broken down in two ways based on the signs of the values. As you have noted they both either need to be positive or both be negative.
The decomposition is a little trickier:
If they are both positive, i.e. $x>4$ then:
$\sqrt{(x-1)(x-4)}=\sqrt{x-1}\cdot\sqrt{x-4}$
And if they are both negative, i.e. $x<1$ then:
$\sqrt{(x-1)(x-4)}=\sqrt{(1-x)(4-x)}=\sqrt{1-x}\cdot\sqrt{4-x}$
Outside of these two ranges one of these radicals isn't real as the value under the radical is negative.