To find the domain of the function \( y = \cos^{-1}(x^2 - 4) \), we need to determine the values of \( x \) for which the expression \( x^2 - 4 \) falls within the range of the inverse cosine function. The range of \( \cos^{-1}(x) \) is defined for values of \( x \) in the interval \([-1, 1]\). Therefore, we need to solve the following inequalities:
1. \( x^2 - 4 \geq -1 \)
2. \( x^2 - 4 \leq 1 \)
### Step 1: Solve the first inequality
Starting with the first inequality:
\[
x^2 - 4 \geq -1
\]
Rearranging gives:
\[
x^2 \geq 3
\]
This can be factored as:
\[
x^2 - 3 \geq 0
\]
Factoring further:
\[
(x - \sqrt{3})(x + \sqrt{3}) \geq 0
\]
To find the critical points, we set \( x - \sqrt{3} = 0 \) and \( x + \sqrt{3} = 0 \), giving us the points \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).
### Step 2: Test intervals for the first inequality
We test the intervals determined by the critical points:
- For \( x < -\sqrt{3} \) (e.g., \( x = -4 \)): \( (-4 - \sqrt{3})(-4 + \sqrt{3}) > 0 \) (positive)
- For \( -\sqrt{3} < x < \sqrt{3} \) (e.g., \( x = 0 \)): \( (0 - \sqrt{3})(0 + \sqrt{3}) < 0 \) (negative)
- For \( x > \sqrt{3} \) (e.g., \( x = 4 \)): \( (4 - \sqrt{3})(4 + \sqrt{3}) > 0 \) (positive)
Thus, the solution to the first inequality is:
\[
x \in (-\infty, -\sqrt{3}] \cup [\sqrt{3}, \infty)
\]
### Step 3: Solve the second inequality
Now, we solve the second inequality:
\[
x^2 - 4 \leq 1
\]
Rearranging gives:
\[
x^2 \leq 5
\]
This can be factored as:
\[
x^2 - 5 \leq 0
\]
Factoring gives:
\[
(x - \sqrt{5})(x + \sqrt{5}) \leq 0
\]
The critical points are \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).
### Step 4: Test intervals for the second inequality
We test the intervals determined by the critical points:
- For \( x < -\sqrt{5} \) (e.g., \( x = -6 \)): \( (-6 - \sqrt{5})(-6 + \sqrt{5}) > 0 \) (positive)
- For \( -\sqrt{5} < x < \sqrt{5} \) (e.g., \( x = 0 \)): \( (0 - \sqrt{5})(0 + \sqrt{5}) < 0 \) (negative)
- For \( x > \sqrt{5} \) (e.g., \( x = 6 \)): \( (6 - \sqrt{5})(6 + \sqrt{5}) > 0 \) (positive)
Thus, the solution to the second inequality is:
\[
x \in [-\sqrt{5}, \sqrt{5}]
\]
### Step 5: Find the intersection of the two solutions
Now, we need to find the intersection of the two sets:
1. From the first inequality: \( (-\infty, -\sqrt{3}] \cup [\sqrt{3}, \infty) \)
2. From the second inequality: \( [-\sqrt{5}, \sqrt{5}] \)
The intersection is:
- For the left part: \( (-\infty, -\sqrt{3}] \cap [-\sqrt{5}, \sqrt{5}] = [-\sqrt{5}, -\sqrt{3}] \)
- For the right part: \( [\sqrt{3}, \infty) \cap [-\sqrt{5}, \sqrt{5}] = [\sqrt{3}, \sqrt{5}] \)
Thus, the final domain of \( y = \cos^{-1}(x^2 - 4) \) is:
\[
x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]
\]
