Step 1 :The function given is \(y=3 \cdot 2^{\sqrt{x}}\).
Step 2 :The base of the exponent, 2, can take any exponent and still be a real number.
Step 3 :However, the exponent is \(\sqrt{x}\), which means x must be greater than or equal to 0, because we cannot take the square root of a negative number in the real number system.
Step 4 :Therefore, the domain of this function is all real numbers greater than or equal to 0.
Step 5 :\(\boxed{x \geq 0}\)