b. Suppose $g$ is a monomial function such that as $x \rightarrow \infty, g(x) \rightarrow \infty$ and as $x \rightarrow-\infty, g(x) \rightarrow \infty$. Which of the following could be a function formula for $f$ ? Select all that apply.
\[
\begin{array}{l}
f(x)=-2 x^{2} \\
f(x)=x^{8} \\
f(x)=\left(\frac{1}{5}\right) x^{4} \\
f(x)=-\left(\frac{1}{2}\right) x^{3} \\
f(x)=x^{5}
\end{array}
\]

Step 1 ：The question is asking for a monomial function that tends to infinity as x tends to both positive and negative infinity. This means that the function must be an even function, because even functions are symmetric about the y-axis and thus have the same behavior as x tends to positive and negative infinity. Therefore, we are looking for a function with an even exponent.

Step 2 ：The functions \(f(x)=-2 x^{2}\), \(f(x)=x^{8}\), and \(f(x)=\left(\frac{1}{5}\right) x^{4}\) are even functions, which means they could be the function g described in the question. However, we also need to check if these functions tend to infinity as x tends to both positive and negative infinity.

Step 3 ：The functions \(f(x)=x^{8}\) and \(f(x)=\left(\frac{1}{5}\right) x^{4}\) are even functions that tend to infinity as x tends to both positive and negative infinity. Therefore, these could be the function g described in the question.

Step 4 ：The possible function formulas for \(f\) are \(f(x)=x^{8}\) and \(f(x)=\left(\frac{1}{5}\right) x^{4}\). So, the final answer is \(\boxed{f(x)=x^{8}, f(x)=\left(\frac{1}{5}\right) x^{4}}\).