Domain of `cos^(-1) [2x^2 - 3]`, where [`**`] denotes the greatest integer function, is equal to

Options

A. `[ 1, sqrt(5/2)]`
B. ` [-sqrt(5/2),-1]`
C. `(-sqrt(5/2),-1] uu [1,sqrt(5/2))`
D. `[- sqrt(5/2) ,-1] uu [1,5 sqrt(1/2)]`

Answer

To find the domain of the function \( f(x) = \cos^{-1} \left[ 2x^2 - 3 \right] \), where \([ \cdot ]\) denotes the greatest integer function, we need to ensure that the expression inside the inverse cosine function is within the valid range for the cosine inverse function.

### Step-by-Step Solution:

1. **Understanding the Range of \( \cos^{-1}(y) \)**:
The function \( \cos^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we need to ensure:
\[ -1 \leq [2x^2 - 3] \leq 1 \]

2. **Breaking Down the Inequalities**:
We can break this down into two separate inequalities:
- \( [2x^2 - 3] \geq -1 \)
- \( [2x^2 - 3] \leq 1 \)

3. **Solving the First Inequality**:
The greatest integer function \( [y] \) gives the largest integer less than or equal to \( y \). Thus, the first inequality \( [2x^2 - 3] \geq -1 \) implies:
\[ 2x^2 - 3 \geq -1 \implies 2x^2 \geq 2 \implies x^2 \geq 1 \implies |x| \geq 1 \]
This gives us two intervals:
\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]

4. **Solving the Second Inequality**:
The second inequality \( [2x^2 - 3] \leq 1 \) implies:
\[ 2x^2 - 3 < 2 \implies 2x^2 < 5 \implies x^2 < \frac{5}{2} \implies |x| < \sqrt{\frac{5}{2}} \approx 1.58 \]
This gives us the interval:
\[ -\sqrt{\frac{5}{2}} < x < \sqrt{\frac{5}{2}} \]

5. **Combining the Intervals**:
Now, we need to combine the results from both inequalities:
- From \( |x| \geq 1 \): \( (-\infty, -1] \cup [1, \infty) \)
- From \( |x| < \sqrt{\frac{5}{2}} \): \( (-\sqrt{\frac{5}{2}}, \sqrt{\frac{5}{2}}) \)

The valid \( x \) values must satisfy both conditions. Thus, we take the intersection:
- For \( x \leq -1 \), the interval is \( (-\sqrt{\frac{5}{2}}, -1] \).
- For \( x \geq 1 \), the interval is \( [1, \sqrt{\frac{5}{2}}) \).

6. **Final Domain**:
Therefore, the domain of the function \( f(x) = \cos^{-1} \left[ 2x^2 - 3 \right] \) is:
\[ \left( -\sqrt{\frac{5}{2}}, -1 \right] \cup [1, \sqrt{\frac{5}{2}}) \]