To determine the range of the function $f(x) = x^3 + 2$, let's analyze it step by step.

Step 1: Consider the function

The function is given as $f(x) = x^3 + 2$.

Step 2: Analyze the cubic term

The term $x^3$ is a cubic function, and cubic functions have the following properties:

• They are continuous.
• They are defined for all real numbers $x$.
• As $x$ approaches $\infty$, $x^3$ approaches $\infty$.
• As $x$ approaches $-\infty$, $x^3$ approaches $-\infty$.

Step 3: Determine the range

Since $x^3$ can take any real value, $x^3 + 2$ will also take any real value, but shifted upwards by 2 units.

Thus, the range of $f(x) = x^3 + 2$ is all real numbers.

Step 4: Select the correct answer

The correct answer from the options provided is "All real numbers".

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How do you find the domain of a function?
2. What is the difference between the range and the codomain of a function?
3. Can you explain why a cubic function's range is all real numbers?
4. How does shifting a function vertically affect its range?
5. What is the impact of adding a constant to a function on its graph?

Tip: Always start analyzing the range by considering the behavior of the base function (in this case, $x^3$) before any transformations are applied.