We know that According to Algebra of Real Functions :
If f and g are two real functions which are defined under the same domain then [tex](\frac{f}{g})(x) = \frac{f(x)}{g(x)}[/tex]
[tex]\implies Given\;Function\;is\;\frac{-1}{x\sqrt{3x - 9}}[/tex]
Now we need find the Domain of this Function :
The Condition for Square Root to be defined is any Expression under it should be Greater than or Equal to Zero.
When Function is a Fraction, it Cannot be defined when the denominator becomes zero. Because when the denominator is zero, the fraction tends to ∞ (because anything divided by zero tends to ∞)
According to Above Conditions Described above, The Given Function is Definable only when the Expression which is under the Square Root is Greater than Zero and x ≠ 0
⇒ 3x - 9 > 0
⇒ 3x > 9
⇒ x > 3
⇒ The Domain of the Given Function is (3 , ∞)
1st Option is the Answer
