Determine whether the function is a polynomial function. If it​ is, state the degree. If it is​ not, tell why not.

x to the power of 2/3 + x^2 + 9

The given function is:

\[
f(x) = x^{2/3} + x^2 + 9
\]

To determine if this is a polynomial function, we need to check if all the exponents of \(x\) are non-negative integers (whole numbers).

1. The term \(x^{2/3}\) has an exponent of \(\frac{2}{3}\), which is **not** a whole number.
2. The term \(x^2\) has an exponent of 2, which **is** a whole number.
3. The constant term \(9\) can be considered as \(9x^0\), where the exponent is \(0\), which is also a whole number.

### Conclusion:
Since the term \(x^{2/3}\) has a fractional exponent, the function is **not** a polynomial. For a function to be a polynomial, all exponents of \(x\) must be non-negative integers.

Would you like more details on polynomial functions or further examples?

Here are 5 related questions to expand on this topic:
1. What defines a polynomial function, and what are some examples of non-polynomial functions?
2. How do you determine the degree of a polynomial function?
3. What is the importance of the degree in polynomial functions?
4. Can a function have negative exponents and still be considered a polynomial?
5. How do polynomial functions behave compared to non-polynomial functions?

**Tip:** Always remember, for a function to be a polynomial, it should only involve whole number exponents of the variable.

Determine whether the function is a polynomial function. If it is, state the degree. If it is not, tell why not. x to the power of 2/3 + x^2 + 9

This guide explains how to determine whether the function x^(2/3) + x^2 + 9 is a polynomial. Explore the concept of polynomial functions, the importance of whole number exponents, and why certain terms like x^(2/3) disqualify the function from being a polynomial. Suitable for Grades 10-12.

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