a. Suppose that $f$ is a monomial function such that as $x \rightarrow \infty, f(x) \rightarrow-\infty$. Which of the following could be a function formula for $f$ ? Select all that apply. \[ \begin{array}{l} f(x)=-x^{9} \\ f(x)=-3 x^{2} \end{array} \] $f(x)=-\left(\frac{1}{7}\right) x^{4}$ $f(x)=x^{10}$ $f(x)=5.5 x^{5}$

Step 1 ：The problem is asking for the function formulas for $f$ that tend to $-\infty$ as $x$ tends to $\infty$. This means we are looking for functions that decrease without bound as $x$ increases.

Step 2 ：For a monomial function $f(x) = ax^n$, where $a$ is the coefficient and $n$ is the degree of the monomial, the function will tend to $-\infty$ as $x$ tends to $\infty$ if and only if $a < 0$ and $n$ is an odd number. This is because an odd degree will preserve the sign of $x$ (i.e., positive $x$ will yield positive result and negative $x$ will yield negative result), and a negative coefficient will flip the sign of the result.

Step 3 ：Therefore, we need to check the sign of the coefficient and the parity of the degree for each function formula.

Step 4 ：Checking the functions $f(x)=-x^{9}$, $f(x)=-3x^{2}$, $f(x)=-\left(\frac{1}{7}\right)x^{4}$, $f(x)=x^{10}$, and $f(x)=5.5x^{5}$, we find that only $f(x)=-x^{9}$ meets the criteria.

Step 5 ：Final Answer: The only function formula for $f$ that could tend to $-\infty$ as $x$ tends to $\infty$ is $f(x)=-x^{9}$. Therefore, the correct answer is \(\boxed{f(x)=-x^{9}}\).