If `f(x)=(1)/(sqrt(1-x^(2)))`, which of the following describes all the real values of x for which f(x) is undefined ?

Options

A. `x=-1` or x = 1
B. `x lt -1` or `x gt 1`
C. `x le - 1` or `x ge 1`
D. `-1 lt x lt 1`

Explanation

To determine the values of \( x \) for which the function \( f(x) = \frac{1}{\sqrt{1 - x^2}} \) is undefined, we need to analyze the denominator of the function.

### Step 1: Identify when the denominator is zero
The function \( f(x) \) is undefined when the denominator is equal to zero. Therefore, we set the denominator to zero:

\[ \sqrt{1 - x^2} = 0 \]

### Step 2: Solve for \( x \)
Squaring both sides gives:

\[ 1 - x^2 = 0 \]

Rearranging this equation, we find:

\[ x^2 = 1 \]

Taking the square root of both sides results in:

\[ x = \pm 1 \]

So, \( x = 1 \) and \( x = -1 \) are points where the function is undefined.

### Step 3: Identify when the expression under the square root is negative
Next, we need to check when the expression under the square root is negative, as the square root of a negative number is also undefined. Thus, we set up the inequality:

\[ 1 - x^2 < 0 \]

Rearranging gives:

\[ -x^2 < -1 \quad \text{or} \quad x^2 > 1 \]

### Step 4: Solve the inequality
Taking the square root of both sides, we have:

\[ |x| > 1 \]

This can be broken down into two cases:

1. \( x > 1 \)
2. \( x < -1 \)

### Step 5: Combine the results
Combining the results from Steps 2 and 4, we find that \( f(x) \) is undefined for:

- \( x = 1 \)
- \( x = -1 \)
- \( x > 1 \)
- \( x < -1 \)

Thus, we can express this as:

\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]

### Final Answer
The values of \( x \) for which \( f(x) \) is undefined are:

\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]