Which of the following functions are polynomial functions? Select all that apply.
\[
\begin{array}{l}
f(x)=4 x^{\frac{1}{3}}+x^{\frac{1}{2}}-3 \\
f(x)=1+x+x^{4} \\
f(x)=x^{2}+x+x^{-1} \\
f(x)=x^{14}-9 \\
f(x)=x^{3}+4 x^{2}+x+13
\end{array}
\]

Step 1 ：A polynomial function is a function that can be expressed in the form of a polynomial. A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). The powers must be nonnegative integers.

Step 2 ：Looking at the given functions, we can see that the first function has fractional powers, so it is not a polynomial function.

Step 3 ：The third function has a negative power, so it is not a polynomial function either.

Step 4 ：The remaining functions all have nonnegative integer powers, so they are polynomial functions.

Step 5 ：The polynomial functions are \(f(x)=1+x+x^{4}\), \(f(x)=x^{14}-9\), and \(f(x)=x^{3}+4 x^{2}+x+13\).

Step 6 ：Final Answer: \(\boxed{f(x)=1+x+x^{4}, f(x)=x^{14}-9, f(x)=x^{3}+4 x^{2}+x+13}\)