Which of the following functions are polynomial functions? Select all that apply.
$f(x)=x^{2}+x+x^{-1}$
$f(x)=4 x^{\frac{1}{3}}+x^{\frac{1}{2}}-4$
$f(x)=x^{17}-5$
$f(x)=1+x+x^{4}$
$f(x)=x^{3}+3 x^{2}+x+14$

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 general form of a polynomial function is \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are constants and \(n\) is a non-negative integer.

Step 2 ：Looking at the given functions, we can see that the first function \(f(x)=x^{2}+x+x^{-1}\) is not a polynomial function because it contains a term with a negative exponent.

Step 3 ：The second function \(f(x)=4 x^{\frac{1}{3}}+x^{\frac{1}{2}}-4\) is also not a polynomial function because it contains terms with fractional exponents.

Step 4 ：The third function \(f(x)=x^{17}-5\), the fourth function \(f(x)=1+x+x^{4}\), and the fifth function \(f(x)=x^{3}+3 x^{2}+x+14\) are all polynomial functions because they only contain terms with non-negative integer exponents.

Step 5 ：The polynomial functions are \(f(x)=x^{17}-5\), \(f(x)=1+x+x^{4}\), and \(f(x)=x^{3}+3 x^{2}+x+14\).

Step 6 ：\(\boxed{f(x)=x^{17}-5, f(x)=1+x+x^{4}, f(x)=x^{3}+3 x^{2}+x+14}\) are the polynomial functions.