\begin{tabular}{lll|} Select the functions that are polynomial functions. \\ \( \begin{array}{lll}2 x^{2}+x-7 & \square x^{-3} & \square 3 x \\ \square \frac{1}{2} x^{3}-2 x^{2}-\sqrt{7} x & \square 3 x^{\frac{1}{2}}-x+4 & \square \frac{1}{2 x+7} \\ \square-x^{\frac{1}{3}}+3 x^{2}+x & 2 x^{2}+5 & 4 x^{7}-2 x^{3}+6 x+9\end{array} \)\end{tabular}

To determine which of the given functions are polynomial functions, we need to recall the definition of a polynomial function. A polynomial function is a function that can be expressed in the form: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants (coefficients), \(n\) is a non-negative integer, and \(x\) is the variable. Importantly, polynomial functions do not include negative exponents, fractional exponents, or any terms involving roots. Now, let's analyze each function provided: 1. \(2 x^{2}+x-7\) - This is a polynomial function (degree 2). 2. \(x^{-3}\) - This is not a polynomial function (negative exponent). 3. \(3 x\) - This is a polynomial function (degree 1). 4. \(\frac{1}{2} x^{3}-2 x^{2}-\sqrt{7} x\) - This is a polynomial function (degree 3). 5. \(3 x^{\frac{1}{2}}-x+4\) - This is not a polynomial function (fractional exponent). 6. \(\frac{1}{2 x+7}\) - This is not a polynomial function (rational function). 7. \(-x^{\frac{1}{3}}+3 x^{2}+x\) - This is not a polynomial function (fractional exponent). 8. \(2 x^{2}+5\) - This is a polynomial function (degree 2). 9. \(4 x^{7}-2 x^{3}+6 x+9\) - This is a polynomial function (degree 7). Now, let's summarize the polynomial functions identified: - \(2 x^{2}+x-7\) - \(3 x\) - \(\frac{1}{2} x^{3}-2 x^{2}-\sqrt{7} x\) - \(2 x^{2}+5\) - \(4 x^{7}-2 x^{3}+6 x+9\) Thus, the polynomial functions from the list are: - \(2 x^{2}+x-7\) - \(3 x\) - \(\frac{1}{2} x^{3}-2 x^{2}-\sqrt{7} x\) - \(2 x^{2}+5\) - \(4 x^{7}-2 x^{3}+6 x+9\)