To determine the values of \( x \) for which the function \( f(x) = \frac{\sqrt{x^2 - 4}}{x - 4} \) is defined, we need to consider two main conditions:
1. The denominator cannot be zero.
2. The expression inside the square root must be non-negative (i.e., \( \sqrt{x^2 - 4} \) must be defined).
### Step 1: Denominator Condition
The denominator \( x - 4 \) cannot be zero. Therefore, we have:
\[
x - 4 \neq 0 \implies x \neq 4
\]
### Step 2: Numerator Condition
Next, we need to ensure that the expression inside the square root is non-negative:
\[
x^2 - 4 \geq 0
\]
This can be factored as:
\[
(x - 2)(x + 2) \geq 0
\]
### Step 3: Finding the Critical Points
The critical points of the inequality \( (x - 2)(x + 2) = 0 \) are \( x = -2 \) and \( x = 2 \). We will analyze the sign of the product \( (x - 2)(x + 2) \) in the intervals determined by these critical points:
- Interval 1: \( (-\infty, -2) \)
- Interval 2: \( (-2, 2) \)
- Interval 3: \( (2, \infty) \)
### Step 4: Testing the Intervals
1. **For \( x < -2 \)** (e.g., \( x = -3 \)):
\[
(-3 - 2)(-3 + 2) = (-5)(-1) = 5 \quad (\text{positive})
\]
2. **For \( -2 < x < 2 \)** (e.g., \( x = 0 \)):
\[
(0 - 2)(0 + 2) = (-2)(2) = -4 \quad (\text{negative})
\]
3. **For \( x > 2 \)** (e.g., \( x = 3 \)):
\[
(3 - 2)(3 + 2) = (1)(5) = 5 \quad (\text{positive})
\]
### Step 5: Conclusion from the Intervals
From the sign analysis, we find that:
- The expression \( (x - 2)(x + 2) \geq 0 \) holds for \( x \in (-\infty, -2] \cup [2, \infty) \).
### Step 6: Combining Conditions
Now, we combine the conditions:
1. \( x \neq 4 \)
2. \( x \in (-\infty, -2] \cup [2, \infty) \)
Thus, the values of \( x \) for which \( f(x) \) is defined are:
\[
x \in (-\infty, -2] \cup [2, \infty) \text{ excluding } 4
\]
### Final Answer
The final solution is:
\[
x \in (-\infty, -2] \cup [2, 4) \cup (4, \infty)
\]
