Final answer:
The range of the function f(x) = x³ - 3x² - 4x is all real numbers. This is because the cubic term causes the function's output to extend to both positive and negative infinity.
Step-by-step explanation:
The question asks for the range of the function f(x) = x³ - 3x² - 4x. To determine this, let's analyze the behavior of the function at extreme values and local extremums. When x approaches positive or negative infinity, f(x) also approaches infinity since the x³ term dominates the equation. This indicates that the function has no bounds as x becomes very large or very small in either direction.
Calculating the derivative, f'(x) = 3x² - 6x - 4, we find the critical points to check for local extremums, but regardless of these, the cubic term will ensure that f(x) will take on all real numbers as its range at some point. Therefore, the range of the function is all real numbers.The correct answer to the question is: 1) All real numbers.