It's wrong answer, the Cardinality of the set A is 12

Let's re-evaluate the problem step by step to identify any mistakes in the initial solution.

Given the function \( f(x) = \frac{\sqrt{x^2 - 36}}{x + 6} \), let's carefully determine the domain of \( f(x) \) by addressing the constraints again.

### Step 1: Analyze the Domain Constraints

#### Condition 1: Non-negativity of the square root argument
\[ x^2 - 36 \geq 0 \]
\[ (x - 6)(x + 6) \geq 0 \]

This inequality holds when:
- \( x \leq -6 \)
- \( x \geq 6 \)

So, \( x \) must be in the intervals \((-\infty, -6] \cup [6, \infty)\).

#### Condition 2: Denominator not equal to zero
\[ x + 6 \neq 0 \]
\[ x \neq -6 \]

### Step 2: Combine the Conditions
From Condition 1, \( x \) must be in \((-\infty, -6] \cup [6, \infty)\).
From Condition 2, \( x \neq -6 \).

Thus, the combined domain of \( f(x) \) is:
\[ x \in (-\infty, -6) \cup [6, \infty) \]

### Step 3: Determine the Integers not in the Domain
We need to identify the set \( A \) of integers that are not in the domain of \( f \).

The integers not in the domain are:
- All integers between \( -5 \) and \( 5 \), inclusive.
- The integer \( -6 \) is specifically excluded.

So, the set \( A \) includes all integers from \(-5\) to \(5\) inclusive, and also excludes \( -6 \).

List the integers from \( -5 \) to \( 5 \):
\[ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \]

### Step 4: Find the Cardinality of the Set \( A \)
The set \( A \) contains the following integers:
\[ A = \{ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \} \]

Counting these integers, we have:
\[ \text{Cardinality of } A = 11 \]

### Re-check for Set Exclusion
- Since \(-6\) is explicitly mentioned, we don't include it in the interval.

Thus, the set \( A \) consists of 11 elements, not 12.

Therefore, the correct cardinality of the set \( A \) is \( 11 \).

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Do you want further details or have any questions? Here are some related questions you might find interesting:

1. How do you determine the domain of a function involving a square root and a fraction?
2. What are the steps to solve inequalities involving quadratic expressions?
3. How do you find the cardinality of a set?
4. What are some examples of functions with restricted domains?
5. How do you solve compound inequalities?

**Tip:** When dealing with functions, always carefully check the interval boundaries and excluded points to ensure the domain is correctly identified.

Explore how to analyze the domain of a function involving square roots and fractions step-by-step. This problem covers key concepts such as functions, domain constraints, and inequalities, with a focus on quadratic inequalities. Suitable for students in Grades 10-12.

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