Answer: The correct option is (D) \(x\geq -7.\)
Step-by-step explanation: We are given to find the domain of the following function:
\(y=3\sqrt{6x+42}~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
The domain of a function y = f(x) is the set of all possible x-values which will make the function defined or will output real y-values.
That is, for the function (i), the domain will be given by all those values of x for which \(3\sqrt{6x+42}\) is real.
Now, the values of y will be real only if the expression under square root is greater than or equal to 0.
Therefore, the domain of function (i) is
\(\sqrt{6x+42}\geq0\\\\\Rightarrow 6x+42\geq 0\\\\\Rightarrow 6x\geq -42\\\\\Rightarrow x\geq -\dfrac{42}{6}\\\\\Rightarrow x\geq -7.\)
Thus, the domain of the given function is \(x\geq -7.\)
Option (D) is CORRECT.
