To determine the range of the function $f(x) = 3x^3$, we need to analyze how the function behaves as $x$ takes different values.

Step-by-step analysis:

1. The function $f(x) = 3x^3$ is a cubic function.
2. A cubic function generally has the following properties:
   • It is continuous and smooth.
   • As $x \to \infty$, $f(x) \to \infty$.
   • As $x \to -\infty$, $f(x) \to -\infty$.

Since the cubic function grows without bound in both positive and negative directions, the range of $f(x)$ is all real numbers.

Conclusion:

The range of the function $f(x) = 3x^3$ is all real numbers, or $\mathbb{R}$.

Let me know if you'd like further clarification or a deeper explanation!

Here are 5 related questions:

1. How do you find the domain of a cubic function like $f(x) = 3x^3$?
2. What is the behavior of the graph of $f(x) = 3x^3$ at the origin?
3. How do transformations (such as scaling) affect the range of cubic functions?
4. How does the range of quadratic functions differ from the range of cubic functions?
5. What happens to the range if we modify the function to $f(x) = 3x^2$?

Tip: For cubic functions, the range is typically all real numbers because they can grow infinitely in both positive and negative directions.