11)
\[
\begin{array}{l}
f(x)=\frac{x}{x^{2}+49} \\
g(x)=\frac{x-4}{x^{2}-16}
\end{array}
\]

Step 1 ：We are given two functions, \(f(x) = \frac{x}{x^{2} + 49}\) and \(g(x) = \frac{x - 4}{x^{2} - 16}\). We are asked to find the domain of these functions.

Step 2 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. For these rational functions, the denominator cannot be zero because division by zero is undefined. So, we need to find the values of x that make the denominator zero and exclude them from the domain.

Step 3 ：For the function \(f(x)\), the denominator is \(x^{2} + 49\). This is always positive for real numbers x, because \(x^{2}\) is always non-negative and we're adding 49 to it. So, the domain of \(f(x)\) is all real numbers.

Step 4 ：For the function \(g(x)\), the denominator is \(x^{2} - 16\). This equals zero when \(x = 4\) or \(x = -4\). So, the domain of \(g(x)\) is all real numbers except 4 and -4.

Step 5 ：\(\boxed{\text{Final Answer: The domain of } f(x) \text{ is all real numbers, and the domain of } g(x) \text{ is all real numbers except } -4 \text{ and } 4. \text{ In interval notation, this is } (- \infty, \infty) \text{ for } f(x) \text{ and } (- \infty, -4) \cup (-4, 4) \cup (4, \infty) \text{ for } g(x).}\)