Problem 7 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The domain of the function \(f(x)=\frac{1}{\sqrt{|x|-x}}\) is (a) \((0, \infty)\) (b) \((-\infty, 0)\) (c) \((-\infty, \infty)-\\{0\\}\) (d) \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The domain of the function is \((0, \infty)\). Choose option (a).

Step 1: Understand the Function's Requirements

To find the domain of the function \( f(x) = \frac{1}{\sqrt{|x|-x}} \), start by identifying conditions that make the expression inside the square root and the fraction valid. The expression inside the square root needs to be non-negative, and the overall expression in the denominator should not be zero.

Step 2: Simplify the Expression \( |x| - x \)

Consider the expression \( |x| - x \). Recall that \(|x| = x\) for \(x \geq 0\) and \(|x| = -x\) for \(x < 0\). For \(x \geq 0\), \(|x| - x = x - x = 0\). For \(x < 0\), \(|x| - x = -x - x = -2x.\) Thus, the expression is zero at \(x = 0\) and negative for \(x < 0\).

Step 3: Apply Validity Conditions

For the square root to be defined, \(|x| - x \geq 0\). As per the simplification, this is true for \(x \geq 0\). However, at \(x = 0\), \(|x| - x = 0\), which makes \(\sqrt{|x|-x}\) defined but the overall expression becomes division by zero for the function. Hence, \(x eq 0\).

Step 4: Determine the Domain

Since the expression \(|x|-x\) is valid and the denominator is not zero only for values \(x > 0\), this implies the domain of the function is \((0, \infty)\).

Step 5: Conclude with the Correct Answer

Out of the given options, the correct choice for the domain of the function \( f(x) = \frac{1}{\sqrt{|x|-x}} \) is (a) \((0, \infty)\).

Key Concepts

Square Root Conditions

In any mathematical expression, particularly those involving square roots, it's important to remember that the quantity inside the square root, known as the radicand, must always be non-negative. This ensures the square root is a real number, as the square root of a negative number is not defined in the realm of real numbers.

For the function we are examining, the expression inside the square root is \(|x|-x\). This means we must solve for conditions where \(|x|-x\) is greater than or equal to zero:

• For \(x \geq 0\), \(|x|-x = 0\). The radicand is zero, which is valid, but not for the entire function.
• For \(x < 0\), simplify to \(|x|-x = -2x\), which results in a negative value since \(-2x\) is negative.

Therefore, square root conditions ensure we're only considering scenarios where the contents under the square root are accessible within the domain of real numbers—specifically non-negative values, leading us to rule out values that make the radicand negative.

Absolute Value Function

The absolute value function, \(|x|\), can sometimes be confusing but it's quite straightforward. It is essentially the distance of a number from zero, regardless of direction on the number line:

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

In the expression \(|x|-x\), which involves both absolute value and regular value, the functionality changes depending on whether \(x\) is positive or negative:

• When \(x \geq 0\), it simplifies to zero \(x - x = 0\).
• When \(x < 0\), it simplifies to \(-2x\).

Understanding how absolute values convert into normal equations based on the sign of \(x\) is crucial in breaking down more complex expressions featuring them effortlessly.

Division by Zero

One of the foundational laws of arithmetic is that division by zero is undefined. Let's break down why that affects the domain of our function:

The function \(f(x) = \frac{1}{\sqrt{|x|-x}}\) brings in an additional constraint with the denominator. If the denominator becomes zero, the function does not exist.

• From step 3, we know \(|x|-x = 0\) at \(x = 0\), which directly causes a division by zero error.
• Thus, this point must be excluded from the domain.

Always remember: if a computation leads to division by zero, that point cannot be a part of the function's domain. Avoid these values to ensure the function remains valid across its defined scope. This understanding is key when determining where the function is realistically calculable without encountering undefined operations.