Problem 164 - A Complete Resource Book in Mathematics for JEE Main

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 \((- fi, 0)\), answer (B).

Step 1: Understanding the Expression Inside the Square Root

To find the domain of the function \(f(x) = \frac{1}{\sqrt{|x| - x}}\), we first focus on the expression inside the square root, \(|x| - x\). For the square root to be defined, \(|x| - x\) must be greater than 0 since the denominator cannot be 0 and square roots of negative numbers are undefined.

Step 2: Analyzing the Absolute Value

The expression \(|x| - x\) changes based on the sign of \(x\). For \(x \geq 0\), \(|x| = x\), so the expression becomes \(x - x = 0\). For \(x < 0\), \(|x| = -x\), so the expression becomes \(-x - x = -2x\). We need \(-2x > 0\).

Step 3: Solving the Inequality

Solving the inequality \(-2x > 0\), we get \(x < 0\). This means the expression \(|x| - x\) is positive when \(x\) is negative. Thus, the function is defined for \(x < 0\).

Step 4: Conclusion on Domain

Given that the inequality holds for \(x < 0\), the domain for the function \(f(x)\) does not include 0 or any positive values. Hence, the correct domain of the function is \((-fi, 0)\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function is a mathematical function that measures the distance of a number from zero on a number line, without considering direction. It's represented as \(|x|\). This means that irrespective of whether the number is positive or negative, it returns the non-negative value.

For example:

• \(|3| = 3\)
• \(|-5| = 5\)

Here, you can see that both \(+3\) and \(-5\) are converted to positive 3 and 5 respectively.

Absolute value expressions often appear in mathematical contexts where the direction or the sign of the number is not significant, particularly in distance and geometric problems.

In the function from our original exercise \(f(x) = \frac{1}{\sqrt{|x| - x}}\), the absolute value is crucial. It changes how we interpret numbers around zero quadrants. This affects whether the expression inside the square root remains positive or not, which determines the domain of the function.

Solving Inequalities Involving Absolute Values

Solving inequalities involving absolute values can be broken down into situations based on the value of the variable.

The expression \(|x|-x\) varies as follows:

• For \(x \geq 0\), the absolute value of \(x\) is \(x\), thus \(|x| - x = 0\).
• For \(x < 0\), the absolute value of \(x\) is \(-x\), leading to \(|x| - x = -2x\).

After establishing these cases, solving the inequality \(-2x > 0\) becomes straightforward. This inequality implies that \(x < 0\).

Why is this important?

Because for the function \(f(x)\), to ensure the square root is defined and finite, the denominator must remain positive.

Understanding this principle is key in finding the domain of such functions.

Finding the Domain of a Function

The domain of a function refers to the complete set of possible input values (x-values) which can produce a valid output in a function.

In simpler words, these are all the x-values where the function is defined and not breaking any mathematical rules, such as dividing by zero or taking the square root of a negative number.

When faced with a function like \(f(x) = \frac{1}{\sqrt{|x| - x}}\):

• We must ensure that what's beneath the square root, \(|x|-x\) remains greater than zero.
• As we solved in the step-by-step, for \(x < 0\), \(|x| - x = -2x\), which turns out positive only if \(x < 0\).

Thus, the domain of this specific function is \((-\infty, 0)\). Understanding domains helps avoid undefined outputs, which can otherwise lead to errors in both fields like mathematics and computational sciences.