A function and its graph are given. Find the domain. (Enter your answer using interval notation.)
f(x) = 

x − 7
x − 8


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.

A function and its graph are given. Find the domain. (Enter your answer using interval notation.) f(x) = (x - 7) / (x - 8)

Explore how to find the domain of the rational function f(x) = (x - 7) / (x - 8). This guide explains the step-by-step process, including key algebraic concepts and the principle of division by zero. Ideal for high school students in Grades 9-11.

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