To solve the problem of finding the range of the inverse function of $f(x) = x^3$, we need to follow these steps:

Step 1: Find the inverse of the function.

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

To find the inverse, swap $x$ and $y$, where $y = x^3$: $x = y^3$ Now, solve for $y$: $y = \sqrt[3]{x}$ Thus, the inverse function is $f^{-1}(x) = \sqrt[3]{x}$.

Step 2: Determine the range of the inverse function.

The cube root function $\sqrt[3]{x}$ is defined for all real numbers. This is because:

• The cube root of any real number (positive, negative, or zero) exists.
• For example, $\sqrt[3]{8} = 2$, $\sqrt[3]{-8} = -2$, and $\sqrt[3]{0} = 0$.

Conclusion:

The range of the inverse function $f^{-1}(x) = \sqrt[3]{x}$ is all real numbers.

Therefore, the correct answer is: All real numbers.

Would you like more details or have any other questions?

Here are 5 related questions for deeper understanding:

1. How do you determine the domain of an inverse function?
2. What are the key properties of cubic functions?
3. How do inverse functions relate to their original functions graphically?
4. Can the range of a function always be used to find the domain of its inverse?
5. What happens if the original function is not one-to-one when finding its inverse?

Tip: Always check if a function is one-to-one (injective) before finding its inverse.