The functions $f$ and $g$ are defined as follows.
\[
\begin{array}{l}
f(x)=\frac{x}{x^{2}+16} \\
g(x)=\frac{x-1}{x^{2}-1}
\end{array}
\]
For each function, find the domain.
Write each answer as an interval or union of intervals.

Step 1 ：The functions $f$ and $g$ are defined as follows. \[ \begin{array}{l} f(x)=\frac{x}{x^{2}+16} \\ g(x)=\frac{x-1}{x^{2}-1} \end{array} \]

Step 2 ：For each function, find the domain. Write each answer as an interval or union of intervals.

Step 3 ：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. The domain of a function is the complete set of possible values of the independent variable. In plain English, this definition means: The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

Step 4 ：For the function $f(x)=\frac{x}{x^{2}+16}$, the denominator $x^{2}+16$ can never be zero for any real number x, because the square of a real number is always non-negative, and adding 16 to it makes it always positive. So, the domain of f is all real numbers.

Step 5 ：For the function $g(x)=\frac{x-1}{x^{2}-1}$, the denominator $x^{2}-1$ can be zero when x = 1 or x = -1. So, the domain of g is all real numbers except 1 and -1.

Step 6 ：Final Answer: The domain of function $f$ is all real numbers, represented as $(-\infty, \infty)$. The domain of function $g$ is all real numbers except -1 and 1, represented as $(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$.