is f(x)=cbrt(x) an even, odd, or neither function?

The function \( f(x) = \sqrt[3]{x} \), or \( f(x) = x^{1/3} \), is an **odd function**. Here's why:

To check whether a function is even, odd, or neither, you apply the following tests:

- **Even function**: \( f(-x) = f(x) \) for all \( x \).
- **Odd function**: \( f(-x) = -f(x) \) for all \( x \).

### Test for oddness:
Let's substitute \( -x \) into the function:

\[
f(-x) = \sqrt[3]{-x} = -\sqrt[3]{x} = -f(x)
\]

Since \( f(-x) = -f(x) \), the function is **odd**.

### Test for evenness:
To be even, \( f(-x) \) would need to equal \( f(x) \), but from the above, we know that \( f(-x) = -f(x) \), so it is not even.

Thus, the function \( f(x) = \sqrt[3]{x} \) is **odd**.

Do you want more details or have any further questions?

### Related questions:
1. What is the graph of \( f(x) = \sqrt[3]{x} \) and how does it confirm that it's odd?
2. Can you give an example of an even function and explain why it's even?
3. How do transformations affect whether a function is even or odd?
4. Is \( f(x) = x^{1/5} \) also an odd function?
5. What is the difference between even, odd, and neither functions?

### Tip:
Odd functions are symmetric about the origin, while even functions are symmetric about the y-axis.

Explore whether the function f(x) = cbrt(x), or f(x) = x^(1/3), is an even or odd function. This detailed explanation involves algebraic concepts, including the test for oddness and the symmetry properties of functions. Perfect for students in Grades 9-11.

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