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To determine where the given function \( f(x) = \frac{11}{x - 12} - 2x \) is continuous, we need to identify where the function might have discontinuities.

### Step-by-Step Analysis:

1. **Identify the Domain of the Function:**
   The function is composed of two parts:
   - \( \frac{11}{x - 12} \): A rational expression that is undefined when the denominator is 0. So, \( x - 12 = 0 \), which gives \( x = 12 \).
   - \( -2x \): This is a linear function, which is continuous everywhere.

2. **Conclusion:**
   The only potential discontinuity arises from the term \( \frac{11}{x - 12} \). The function is undefined at \( x = 12 \), meaning the function will be discontinuous at this point.

3. **Interval Notation:**
   The function is continuous for all \( x \) except \( x = 12 \). In interval notation, this is expressed as:
   \[
   (-\infty, 12) \cup (12, \infty)
   \]

Thus, the function is continuous on \( (-\infty, 12) \cup (12, \infty) \).

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Do you want more details or have any questions?

Here are five related questions to deepen your understanding:

1. What happens to the value of \( f(x) \) as \( x \) approaches 12 from the left and right?
2. How would you describe a removable versus a non-removable discontinuity?
3. Can a function be continuous at every point except for a single value?
4. How do rational functions like \( \frac{11}{x - 12} \) behave near their vertical asymptotes?
5. What are some other examples of functions that are continuous everywhere except for specific points?

**Tip:** Whenever you encounter a rational function, always check the denominator to find potential points of discontinuity.

At what points is the given function f(x) = 11 / (x - 12) - 2x continuous? Describe the set of x-values where the function is continuous, using interval notation.

Explore how to find the continuity of the function f(x) = 11 / (x - 12) - 2x. This step-by-step solution identifies the point of discontinuity and provides the interval notation for where the function is continuous. Suitable for students in Grades 11-12.

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