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