To find the domain of the function

$f(x) = \frac{x - 7}{x - 8}$

we need to identify the values of $x$ for which the function is defined. This function is a rational expression, and it is undefined wherever the denominator equals zero, as division by zero is undefined in mathematics.

Steps:

1. Set the denominator equal to zero and solve for $x$: $x - 8 = 0$ $x = 8$

2. Since $f(x)$ is undefined when $x = 8$, we exclude $x = 8$ from the domain.

3. Therefore, the domain of $f(x)$ includes all real numbers except $x = 8$.

Domain in Interval Notation:

$(-\infty, 8) \cup (8, \infty)$

Let me know if you want further details, or if you have any questions!

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Here are some related questions to deepen understanding:

1. How would the domain change if the denominator were $x - 8 + 1$?
2. What are the domain and range of $f(x) = \frac{1}{x}$?
3. How do you determine the range of a function like $f(x) = \frac{x - 7}{x - 8}$?
4. What does the graph of $f(x) = \frac{x - 7}{x - 8}$ look like around $x = 8$?
5. How does a vertical asymptote relate to the domain of a function?

Tip: To find the domain of a rational function, always look for values that make the denominator zero, as these are the values excluded from the domain.