For each of the following rational functions, determine the monomial function that best approximates the end behavior of the given function. a. As $x \rightarrow \pm \infty$, the function $f(x)=\frac{x^{4}}{x^{2}+10 x+3}$ behaves like:

Step 1 ：First, we need to understand the meaning of the problem. The problem is asking us to find the monomial function that best approximates the end behavior of the given function as $x$ approaches positive or negative infinity.

Step 2 ：The given function is $f(x)=\frac{x^{4}}{x^{2}+10 x+3}$. We can see that the degree of the numerator is higher than the degree of the denominator. This means that as $x$ approaches infinity, the numerator will grow faster than the denominator.

Step 3 ：To find the monomial function that best approximates the end behavior, we can divide each term in the numerator and the denominator by $x^{2}$, the highest power of $x$ in the denominator. This gives us $f(x)=\frac{x^{2}}{1+\frac{10}{x}+\frac{3}{x^{2}}}$.

Step 4 ：As $x$ approaches infinity, the terms $\frac{10}{x}$ and $\frac{3}{x^{2}}$ will approach 0. Therefore, the function $f(x)$ behaves like $f(x)=x^{2}$ as $x$ approaches infinity.

Step 5 ：Similarly, as $x$ approaches negative infinity, the terms $\frac{10}{x}$ and $\frac{3}{x^{2}}$ will also approach 0. Therefore, the function $f(x)$ behaves like $f(x)=x^{2}$ as $x$ approaches negative infinity.

Step 6 ：Therefore, the monomial function that best approximates the end behavior of the given function as $x$ approaches positive or negative infinity is $f(x)=x^{2}$.

Step 7 ：Finally, we need to check whether our results meet the requirements of the problem. The problem asks for the monomial function that best approximates the end behavior of the given function. Our result, $f(x)=x^{2}$, is a monomial function, and it approximates the end behavior of the given function as $x$ approaches positive or negative infinity. Therefore, our result meets the requirements of the problem.

Step 8 ：The final answer is $\boxed{x^{2}}$.