Domain of the function `(sqrt(1+x)-sqrt(1-x))/(x )` is

Options

A. `(-1,1)`
B. `(-1,1)- {0}`
C. `[-1,1]`
D. `[-1,1]-{0}`

Answer

To find the domain of the function \( f(x) = \frac{\sqrt{1+x} - \sqrt{1-x}}{x} \), we need to ensure that the expression is defined and valid. This involves checking the conditions for the square roots and ensuring the denominator is not zero.

### Step 1: Conditions for the square roots
The expression contains two square roots: \( \sqrt{1+x} \) and \( \sqrt{1-x} \). For these square roots to be defined, their arguments must be non-negative.

1. **Condition for \( \sqrt{1+x} \)**:
\[ 1+x \geq 0 \implies x \geq -1 \]

2. **Condition for \( \sqrt{1-x} \)**:
\[ 1-x \geq 0 \implies x \leq 1 \]

### Step 2: Condition for the denominator
The denominator of the function is \( x \). For the function to be defined, \( x \) cannot be zero:
\[ x \neq 0 \]

### Step 3: Combine the conditions
Now we have three conditions:
1. \( x \geq -1 \)
2. \( x \leq 1 \)
3. \( x \neq 0 \)

### Step 4: Determine the domain
The first two conditions \( x \geq -1 \) and \( x \leq 1 \) combine to give the interval:
\[ -1 \leq x \leq 1 \]
However, we must exclude \( x = 0 \) from this interval. Therefore, the domain of the function can be expressed as:
\[ [-1, 0) \cup (0, 1] \]

### Final Answer
The domain of the function \( f(x) = \frac{\sqrt{1+x} - \sqrt{1-x}}{x} \) is:
\[ [-1, 0) \cup (0, 1] \]

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