Final answer:
The transition from f(x) to g(x) involves a horizontal shift, a vertical stretch, and a downward translation, but the cubic end behavior remains the same: decreasing as x approaches negative infinity and increasing as x approaches positive infinity.
Step-by-step explanation:
The transition from f(x) = x^3 to g(x) = 3(x-2)^3 - 8 consists of several transformations. The function is first translated horizontally 2 units to the right to get (x-2)^3. Then, it is vertically stretched by a factor of 3, and finally, it is translated 8 units downward. These transformations neither change the end behavior of the function nor its overall cubic shape, which means the end behavior remains the same as the parent function f(x) = x^3.
For the cubic function f(x) = x^3, as x goes to negative infinity, the function decreases without bound, and as x goes to positive infinity, the function increases without bound. The transformed function g(x) = 3(x-2)^3 - 8 maintains this behavior. Hence, the end behavior of g(x) is: as x approaches negative infinity, g(x) decreases without bound (c), and as x approaches positive infinity, g(x) increases without bound (d).