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. \[ \begin{array}{l} f(x)=-\left(\frac{1}{7}\right) x^{4} \\ f(x)=-3 x^{2} \\ f(x)=5.5 x^{5} \\ f(x)=x^{10} \\ f(x)=-x^{9} \end{array} \] 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. \[ \begin{array}{l} \square(x)=-2 x^{2} \\ \square(x)=x^{8} \\ \square f(x)=\left(\frac{1}{5}\right) x^{4} \\ \square f(x)=-\left(\frac{1}{2}\right) x^{3} \\ \square f(x)=x^{5} \end{array} \]
Solution
Step 1 ：Given that $f$ is a monomial function such that as $x \rightarrow \infty, f(x) \rightarrow -\infty$. This implies that the function must have an odd degree and a negative coefficient.
Step 2 ：By substituting a large value of $x$ into each function, we can determine which functions tend towards negative infinity as $x$ tends towards infinity.
Step 3 ：For $f(x)=-\left(\frac{1}{7}\right) x^{4}$, $f(x)=-3 x^{2}$, and $f(x)=-x^{9}$, the results are negative for a large $x$, which means they tend towards negative infinity as $x$ tends towards infinity.
Step 4 ：For $f(x)=5.5 x^{5}$ and $f(x)=x^{10}$, the results are positive for a large $x$, which means they tend towards positive infinity as $x$ tends towards infinity.
Step 5 ：Therefore, the function formulas for $f$ could be $f(x)=-\left(\frac{1}{7}\right) x^{4}$, $f(x)=-3 x^{2}$, and $f(x)=-x^{9}$.
Step 6 ：Final Answer: \(\boxed{f(x)=-\left(\frac{1}{7}\right) x^{4}, f(x)=-3 x^{2}, f(x)=-x^{9}}\)
