Range of the function
`f(x)=(cos^(-1)abs(1-x^(2)))` is

Options

A. `[0,pi/2]`
B. `[0,pi/3]`
C. `(o,pi)`
D. `(pi/2,pi)`

Answer

To find the range of the function \( f(x) = \cos^{-1}(|1 - x^2|) \), we will follow these steps:

### Step 1: Determine the domain of the function
The function \( \cos^{-1}(y) \) is defined for \( y \) in the interval \([-1, 1]\). Therefore, we need to ensure that \( |1 - x^2| \) lies within this interval.

### Step 2: Analyze the expression \( |1 - x^2| \)
We can break this down into two cases based on the value of \( 1 - x^2 \):

1. **Case 1**: \( 1 - x^2 \geq 0 \) (which implies \( x^2 \leq 1 \) or \( -1 \leq x \leq 1 \))
- Here, \( |1 - x^2| = 1 - x^2 \).
- The values of \( 1 - x^2 \) range from \( 1 \) (when \( x = 0 \)) to \( 0 \) (when \( x = \pm 1 \)).

2. **Case 2**: \( 1 - x^2 < 0 \) (which implies \( x^2 > 1 \) or \( x < -1 \) or \( x > 1 \))
- Here, \( |1 - x^2| = x^2 - 1 \).
- The values of \( x^2 - 1 \) start from \( 0 \) (when \( x = \pm 1 \)) and go to \( \infty \) as \( |x| \) increases.

### Step 3: Find the range of \( |1 - x^2| \)
From the analysis:
- In **Case 1**, \( |1 - x^2| \) ranges from \( 0 \) to \( 1 \).
- In **Case 2**, \( |1 - x^2| \) starts from \( 0 \) and goes to \( \infty \).

However, since \( \cos^{-1}(y) \) is only defined for \( y \) in \([-1, 1]\), we restrict our consideration to the interval \( [0, 1] \).

### Step 4: Determine the range of \( f(x) = \cos^{-1}(|1 - x^2|) \)
- When \( |1 - x^2| = 0 \) (at \( x = \pm 1 \)), \( f(x) = \cos^{-1}(0) = \frac{\pi}{2} \).
- When \( |1 - x^2| = 1 \) (at \( x = 0 \)), \( f(x) = \cos^{-1}(1) = 0 \).

Thus, as \( |1 - x^2| \) varies from \( 0 \) to \( 1 \), \( f(x) \) will take values from \( \frac{\pi}{2} \) to \( 0 \).

### Step 5: Final range of the function
The range of the function \( f(x) = \cos^{-1}(|1 - x^2|) \) is:
\[ \text{Range} = [0, \frac{\pi}{2}] \]