Functions and their domains

I was looking over my notes recently, and this kind of problem was written

$\displaystyle f(x)=\sqrt{5-3x}$

The domain is defined for all values of x<(inclusive)$\displaystyle \frac{5}{3}$, right?

then I have something weird written for function

$\displaystyle \frac{1}{\sqrt{5-3x}} $

the domain is defined for x belongs to $\displaystyle ]-\infty,\frac{5}{3}]\cup $x belongs to $\displaystyle [\frac {5}{3},\infty[$

This can't be right?

I clearly rememver myself saying that the answer is x is greater than 5 over three, and the teacher said it was correct.