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.