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.
$f(x)=-\left(\frac{1}{2}\right) x^{3}$
$f(x)=x^{8}$
$f(x)=-2 x^{2}$
$f(x)=x^{5}$
$f(x)=\left(\frac{1}{5}\right) x^{4}$

Step 1 ：The question is asking for the function formulas for $f$ that satisfy the conditions of $g$. A monomial function is a function of the form $f(x) = ax^n$ where $a$ is a constant and $n$ is a nonnegative integer.

Step 2 ：The behavior of the function as $x$ approaches infinity or negative infinity depends on the degree $n$ and the sign of $a$. If $n$ is even, the function approaches infinity as $x$ approaches both infinity and negative infinity if $a$ is positive, and it approaches negative infinity in both cases if $a$ is negative.

Step 3 ：If $n$ is odd, the function approaches infinity as $x$ approaches infinity and negative infinity as $x$ approaches negative infinity if $a$ is positive, and the opposite if $a$ is negative.

Step 4 ：Therefore, the function formulas for $f$ that could satisfy the conditions of $g$ are those where $n$ is even and $a$ is positive, or where $n$ is odd and $a$ is negative.

Step 5 ：The function formulas for $f$ that could satisfy the conditions of $g$ are $f(x)=-\left(\frac{1}{2}\right) x^{3}$, $f(x)=x^{8}$, and $f(x)=\left(\frac{1}{5}\right) x^{4}$. \(\boxed{f(x)=-\left(\frac{1}{2}\right) x^{3}, f(x)=x^{8}, f(x)=\left(\frac{1}{5}\right) x^{4}}\)