RATIONAL EXPRESSIONS
Domain of a rational function: Interval notation
The functions $f$ and $g$ are defined as follows.
\[
\begin{array}{l}
f(x)=\frac{x-5}{x^{2}-14 x+45} \\
g(x)=\frac{x^{2}}{x-7}
\end{array}
\]
For each function, find the domain.
Write each answer as an interval or union of intervals.

Step 1 ：Define the functions $f(x)=\frac{x-5}{x^{2}-14 x+45}$ and $g(x)=\frac{x^{2}}{x-7}$

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 a rational function, the domain is all real numbers except those that make the denominator equal to zero, because division by zero is undefined.

Step 3 ：To find the domain of the functions $f(x)$ and $g(x)$, we need to find the values of x that make the denominator equal to zero and exclude them from the domain.

Step 4 ：For $f(x)$, we need to solve the equation $x^{2}-14x+45=0$. The solutions are 5 and 9. So, the domain of $f(x)$ is all real numbers except 5 and 9. In interval notation, this is $(-\infty, 5) \cup (5, 9) \cup (9, \infty)$.

Step 5 ：For $g(x)$, we need to solve the equation $x-7=0$. The solution is 7. So, the domain of $g(x)$ is all real numbers except 7. In interval notation, this is $(-\infty, 7) \cup (7, \infty)$.

Step 6 ：\(\boxed{\text{The domain of } f(x) \text{ is } (-\infty, 5) \cup (5, 9) \cup (9, \infty) \text{ and the domain of } g(x) \text{ is } (-\infty, 7) \cup (7, \infty)}\)