In Problems 1-4, determine whether the given function is a polynomial function. \( \begin{array}{ll}\text { 1. } f(x)=x^{2}-x^{4}+4 & \text { 2. } f(x)=\frac{x^{3}+7 x-3}{3} \\ \text { 3. } g(x)=\frac{1}{x^{2}+2 x+1} & \text { 4. } g(x)=2^{-3} x^{3}\end{array} \)

**1. \( f(x)=x^2-x^4+4 \)** All terms in the function are monomials with nonnegative integer exponents and constant coefficients. Therefore, \[ f(x)=x^2-x^4+4 \] is a polynomial. **2. \( f(x)=\frac{x^3+7x-3}{3} \)** Since division by a nonzero constant preserves the polynomial structure, \[ f(x)=\frac{1}{3}(x^3+7x-3) \] is a polynomial. **3. \( g(x)=\frac{1}{x^2+2x+1} \)** The denominator can be factored as: \[ x^2+2x+1=(x+1)^2. \] Thus, \[ g(x)=\frac{1}{(x+1)^2}. \] This is not a polynomial because it involves taking the reciprocal of a polynomial expression; the result is a rational function in which the denominator is not \(1\). **4. \( g(x)=2^{-3}x^3 \)** Notice that \(2^{-3}=\frac{1}{8}\), so \[ g(x)=\frac{1}{8}x^3. \] This is a monomial with a nonnegative integer exponent and a constant factor. Hence, it is a polynomial.