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.