Final answer:
The domain of the function y = 3 tan(2/3 x) is all real numbers except odd integer multiples of 3π/4, because the tangent function is undefined for those values.
Step-by-step explanation:
The function in question is y = 3 tan(2/3 x). To find the domain of this function, we must identify the values of x for which the function is undefined. The tangent function is undefined when its argument is an odd multiple of π/2. In this case, the argument of the tangent function is 2/3 x. This means we need to find the values of x that make 2/3 x equal to an odd multiple of π/2.
We can set up the equation 2/3 x = (2n+1) π/2, where n is an integer. Solving for x gives x = (3/2)(2n+1) π. This simplifies to x = (3n+3/2) π. Thus, the function is undefined for these values of x, which are odd multiples of 3π/4.
The correct domain of the function y = 3 tan(2/3 x) is option 1: all real numbers except odd integer multiples of 3π/4.