Is the notation of the domain for $\displaystyle
\frac{(10)\!\cdot\!\ln(x)}{x}$
correct?:
D=R\(-infinity;0]
If the domain was: not 0, and nothing from -infinity to and with -2:
D=R\{0};(-infinity;-2]?
many thanks
The domain of ln is from (0, infinity) and domain of 1/x is $\displaystyle x \neq 0$
In order to determine the domain, we must compare the two possibilities. Due to ln, we know we can't have any negative values.
Therefore, we are left with (0,infinity).
But we could also use the rules of logs to say the domain is $\displaystyle x \neq 0$.
If we do this $\displaystyle \frac{ln(x^{10})}{x}$, then the domain changes.
As others have stated, the usual way to write the domain would be $\displaystyle (0,\infty)$.
Technically what you gave, $\displaystyle \mathbb{R} \setminus (-\infty,0] $, is also correct, but it's a bit awkward because it's not as clear as the usual way.
Your set notation is correct, though.
Actually, I didn't look very closely at your notation. To clarify, you should have used a comma, not a semicolon.
Also, for the second hypothetical domain, it should be either:$\displaystyle \mathbb{R} \setminus (\{0\} \cup (-\infty,-2])$or$\displaystyle \mathbb{R} \setminus \{0\} \setminus (-\infty,-2]$However, the usual way you would write such a domain would be like so:$\displaystyle (-2,0) \cup (0,\infty)$