https://r2.erweima.ai/mathbot/image/17263896910098kmgoved.jpg

The problem asks for the domain of the function \( f(x) = \frac{6}{x - 7} \).

### Step-by-Step Analysis:
1. **Identify the Domain Restrictions:**
   The function \( f(x) = \frac{6}{x - 7} \) involves a fraction. A fraction is undefined when its denominator is zero. Therefore, we must find when \( x - 7 = 0 \).
   
   Solving \( x - 7 = 0 \):
   \[
   x = 7
   \]
   So, \( f(x) \) is undefined when \( x = 7 \).

2. **Determine the Domain:**
   The function is defined for all real numbers except \( x = 7 \). Therefore, the domain is all real numbers except 7.

   In interval notation, the domain is:
   \[
   (-\infty, 7) \cup (7, +\infty)
   \]

3. **Review the Choices:**
   - \( \{ x \mid x \neq 7 \} \): This is the correct option because it says the function is defined for all values except \( x = 7 \).
   - \( \{ x \mid x > 7 \} \): This is incorrect since it excludes all values less than 7, which are part of the domain.
   - \( \{ x \mid x < 7 \} \): This is also incorrect because it excludes values greater than 7, which are part of the domain.

4. **Interval Notation:**
   The correct interval notation for the domain is:
   \[
   (-\infty, 7) \cup (7, +\infty)
   \]
   Entering just \( (5) \) as shown in the image is incorrect because 5 is just a point, not an interval.

Would you like more details or clarification on any part? 

Here are 5 related questions to explore:
1. How do you find the domain of a function involving square roots?
2. What happens to the domain when you have a function with multiple fractions?
3. How do you find the range of a rational function?
4. Can the domain of a polynomial function ever be restricted?
5. What is the domain of \( f(x) = \frac{1}{x^2 - 1} \)?

**Tip:** Always check for values that make the denominator zero when finding the domain of rational functions!

Find the domain of the function f(x) = 6/(x - 7).

Learn how to find the domain of the rational function f(x) = 6/(x - 7). Understand why the function is undefined at x = 7 and how to express the domain in interval notation. This is a key concept in algebra, suitable for students in Grades 9-12.

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