The domain of `y=cos^(-1)(x^2-4)` is :

Options

A. [3,5]
B. `[0,pi]`
C. `[-sqrt5,-sqrt3] nn [- sqrt5, sqrt3]`
D. `[-sqrt5, - sqrt 3] uu [ sqrt3, sqrt5]`

Answer

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 \( x^2 - 4 \) lies within the range of the inverse cosine function, which is defined for values between -1 and 1, inclusive.

### Step-by-step Solution:

1. **Set up the inequality**:
The expression inside the inverse cosine must satisfy:
\[ -1 \leq x^2 - 4 \leq 1 \]

2. **Break it into two inequalities**:
We can split this compound inequality into two separate inequalities:
\[ x^2 - 4 \geq -1 \quad \text{and} \quad x^2 - 4 \leq 1 \]

3. **Solve the first inequality**:
For the first inequality \( x^2 - 4 \geq -1 \):
\[ x^2 - 4 + 4 \geq -1 + 4 \implies x^2 \geq 3 \]
This implies:
\[ x \leq -\sqrt{3} \quad \text{or} \quad x \geq \sqrt{3} \]

4. **Solve the second inequality**:
For the second inequality \( x^2 - 4 \leq 1 \):
\[ x^2 - 4 \leq 1 \implies x^2 \leq 5 \]
This implies:
\[ -\sqrt{5} \leq x \leq \sqrt{5} \]

5. **Combine the results**:
Now we need to combine the results from both inequalities:
- From \( x^2 \geq 3 \): \( x \leq -\sqrt{3} \) or \( x \geq \sqrt{3} \)
- From \( x^2 \leq 5 \): \( -\sqrt{5} \leq x \leq \sqrt{5} \)

The valid intervals for \( x \) that satisfy both conditions are:
\[ x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \]

### Final Domain:
Thus, the domain of \( y = \cos^{-1}(x^2 - 4) \) is:
\[ x \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}] \]