Problem 48 - Calculus for JEE ADVANCED

Question

\(f(x)=\sqrt{|x|-\\{x\\}}\) is defined if \(|x| \geq\\{x\\}\) or \(\quad x \in\left(-\infty-\frac{1}{2}\right] \cup[0, \infty)\) or \(Y \in[0, \infty)\)

Step-by-Step Solution

Answer

The function is defined for \(x \in (-\infty, -\frac{1}{2}] \cup [0, \infty)\) when \(Y \geq 0\).

Step 1: Understanding the Function

The given function is defined as \(f(x) = \sqrt{|x| - \{x\}}\). The expression inside the square root, \(|x| - \{x\}\), must be non-negative for the square root to be real. We need to analyze when this condition holds true.

Step 2: Analyzing the Modulus

The modulus \(|x|\) is the absolute value of \(x\). We consider two cases: \(x \geq 0\) (then \(|x| = x\)) and \(x < 0\) (then \(|x| = -x\)). This helps us examine the expression \(|x| - \{x\}\).

Step 3: Understanding the Fractional Part

The function \(\{x\}\) denotes the fractional part of \(x\), which is \(x - \lfloor x \rfloor\). It ranges from \(0\) to \(1\) (not inclusive of \(1\)).

Step 4: Determining When the Expression is Non-negative

For the expression to be non-negative, analyze the inequality \(|x| \geq \{x\}\). If \(x\) is non-negative (\(|x| = x\)), the inequality becomes \(x \geq x - \lfloor x \rfloor\), which is always true for any integer or non-negative number \(x\). If \(x\) is negative (\(|x| = -x\)), then we simplify the inequality as \(-x \geq x - \lfloor x \rfloor\), ensuring no real values unless it's narrowly defined.

Step 5: Interpreting the Given Set

The problem states three possible conditions for the domain: \(|x| \geq \{x\}\), \(x \in (-\infty, -\frac{1}{2}] \cup [0, \infty)\), or \(Y \geq 0\). The condition \(x \in (-\infty, -\frac{1}{2}] \cup [0, \infty)\) aligns with making the expression non-negative largely because \(-\frac{1}{2}\) as a boundary ensures that \(-x > \{x\}\) can't be satisfied easily in the negative realm unless it's paired to these bounds cleverly.

Step 6: Final Interpretation of Conditions

The correct interpretation combines our insights from above: indeed, \(|x| \geq \{x\}\) is satisfied on viable numbers meeting \(x \in (-\infty, -\frac{1}{2}] \cup [0, \infty)\), securing that the domain only includes expressions evaluated non-negatively, allowing \(Y\) to reflect non-negative real calculations.

Key Concepts

Absolute Value

The absolute value of a number, represented as \(|x|\), is a fundamental concept in mathematics, and it refers to the non-negative value of a number without regard to its sign. In simpler terms, regardless of whether \(x\) is positive or negative, the absolute value \(|x|\) is always positive. Here’s how it works:

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

For example, if you have \(x = 4\), then \(|x| = 4\). Conversely, if \(x = -4\), then \(|-4| = 4\), as the absolute value negates the negative sign. By looking at the absolute value, you're simply looking at the "distance" from zero on the number line, which is why it's always positive. This feature of the absolute value is crucial when analyzing inequalities and functions that involve absolute values like in the given function \(f(x) = \sqrt{|x| - \{x\}}\).

Square Root Functions

Square root functions involve the square root of a quantity. The square root of a number \(a\) is a value that, when multiplied by itself, gives the number \(a\). Mathematically, it is expressed as \(\sqrt{a}\). For the function \(f(x) = \sqrt{|x| - \{x\}}\),we are looking for when the expression inside the square root \(|x| - \{x\}\) is non-negative.This is crucial because a square root function is only defined for non-negative numbers in the real number system. Hence, the expression inside must be zero or positive.

• If \(|x|\) is greater than or equal to \(\{x\}\), the square root function can accept the value, ensuring \(f(x)\) is real.
• This means if you find a scenario where \(|x| < \{x\}\), it becomes undefined or imaginary, which is generally not acceptable for real numbers.

Thus, exploring when this holds true plays a crucial role in defining the domain of square root functions.

Inequalities in Functions

Inequalities are used to define a range or domain for which a function is valid. In the problem at hand, the inequality \(|x| \geq \{x\}\) determines when the function \(f(x) = \sqrt{|x| - \{x\}}\) is defined. What this inequality is asking is for the absolute value of \(x\) to be larger than or equal to its fractional part \(\{x\}\).
The fractional part \(\{x\}\) is essentially the non-integer part of \(x\), calculated as \(x - \lfloor x \rfloor\), and it lies between 0 (inclusive) and 1 (exclusive). Exploring this inequality helps establish boundaries for \(x\):

• For non-negative values of \(x\), this inequality generally holds, making the square root function defined over degrees in this range.
• When \(x\) is negative, additional care must be taken to ensure \(|x|\) sufficiently exceeds \(\{x\}\), typically resulting in specific allowable intervals.

Understanding how these intervals are determined by the inequality changes what real \(x\) values can produce valid and meaningful results from the function, affirming its importance in analyzing mathematical functions.