Question

Domain of f(x) = \(\frac{x}{|-|x|}\) is

• A. R - [-1, 1]
• B. (-∞, 1)
• C. (-∞, 1) ∪ (0, 1)
• D. R - {-1, 1}

Answer 1

The Correct Option is D

We are asked to find the domain of the function \( f(x) = \frac{x}{1 - |x|} \).

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

For a rational function (a fraction), the function is undefined when the denominator is equal to zero. 

In this case, the denominator is \( 1 - |x| \).

We set the denominator equal to zero and solve for x:

\[ 1 - |x| = 0 \]

Add \( |x| \) to both sides:

\[ 1 = |x| \]

The equation \( |x| = 1 \) means that x can be either 1 or -1.

\[ x = 1 \quad \text{or} \quad x = -1 \]

These are the values of x for which the function \( f(x) \) is undefined.

Therefore, the domain of the function is the set of all real numbers except for -1 and 1.

In set notation, this is written as \( \mathbb{R} - \{-1, 1\} \) or \( \mathbb{R} \setminus \{-1, 1\} \).

In interval notation, the domain is \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

Comparing this with the given options:

• R - [-1, 1] (Incorrect, excludes the interval between -1 and 1)
• (-∞, 1) (Incorrect, excludes -1 and values greater than 1)
• (-∞, 1) ∪ (0, 1) (Incorrect, simplifies to (-∞, 1))
• R - {-1, 1} (Correct)

The correct domain is R - {-1, 1}.

Answer 2

The function \( f(x) = \frac{x}{|x|} \) involves division by the absolute value of \( x \).
The domain of this function excludes \( x = 0 \), as division by zero is undefined.
Therefore, the domain is all real numbers except \( x = 0 \), or \(\mathbb{R} \setminus \{0\}\).
The function \( f(x) \) is defined for all values except \( x = 1 \) and \( x = -1 \), where it will result in an indeterminate form.

The correct answer is (D) : R - {-1, 1}