Step 1 :A monomial function is a function of the form \(f(x) = ax^n\), where \(a\) is a constant and \(n\) is a nonnegative integer. The function \(f(x)\) tends to \(-\infty\) as \(x \rightarrow \infty\) if and only if \(a < 0\) and \(n\) is an odd integer. This is because if \(n\) is an even integer, then \(f(x)\) will tend to \(+\infty\) as \(x \rightarrow \infty\) regardless of the sign of \(a\). Therefore, we need to find the functions among the given options that satisfy these conditions.
Step 2 :The functions that satisfy the conditions are \(f(x)=-3 x^{2}\), \(f(x)=-x^{9}\), and \(f(x)=-\left(\frac{1}{7}\right) x^{4}\). These are the functions for which the function value decreases as \(x\) increases, and the function value is negative for all \(x > 0\). Therefore, these are the functions for which \(f(x) \rightarrow -\infty\) as \(x \rightarrow \infty\).
Step 3 :Final Answer: The function formulas for \(f\) could be \(f(x)=-3 x^{2}\), \(f(x)=-x^{9}\), and \(f(x)=-\left(\frac{1}{7}\right) x^{4}\).