Problem 95 - A Complete Resource Book in Mathematics for JEE Main

Question

The domain of the function \(f(x)=\ln \left\\{\operatorname{sgn}\left(9-x^{2}\right)\right\\}+\sqrt{[x]^{3}-4[x]}\), where \([\cdot]\) denotes integral part, is (A) \((-2,1) \cup(2,3)\) (B) \([-2,1) \cup[2,3)\) (C) \([-2,1] \cup[2,3)\) (D) \([-2,1) \cup[2,3]\)

Step-by-Step Solution

Answer

The domain is \([-2,1) \cup [2,3)\).

Step 1: Identify restrictions for the natural logarithm

The function contains a natural logarithm: \( \ln(\operatorname{sgn}(9-x^2)) \). The logarithm is only defined for positive arguments, so we need \( \operatorname{sgn}(9-x^2) > 0 \). This means \( 9-x^2 > 0 \), which simplifies to \( x^2 < 9 \). Thus, \( -3 < x < 3 \).

Step 2: Identify restrictions for the square root function

The square root function \( \sqrt{[x]^3 - 4[x]} \) is only defined for non-negative arguments, so \( [x]^3 - 4[x] \geq 0 \). The behavior depends on the value of \([x]\). We analyze this for integer \([x]\) values within the domain from Step 1.

Step 3: Analyze each integer interval within the domain

Within \( -3 < x < 3 \), consider integer intervals: \([-3,-2)\), \([-2,-1)\), \([-1,0)\), \([0,1)\), \([1,2)\), \([2,3)\):- For \([-3,-2)\), \([x] = -3\), \(-27 - 4(-3) = -15\) (negative, not included).- For \([-2,-1)\), \([x] = -2\), \(-8 - 4(-2) = 0\) (acceptable, included).- For \([-1,0)\), \([x] = -1\), \(-1 - 4(-1) = 3\) (acceptable, included).- For \([0,1)\), \([x] = 0\), \(0 - 0 = 0\) (acceptable, included).- For \([1,2)\), \([x] = 1\), \(1 - 4(1) = -3\) (negative, not included).- For \([2,3)\), \([x] = 2\), \(8 - 4(2) = 0\) (acceptable, included).Considering these results: \([-2,1)\cup [2,3)\).

Step 4: Combine the results

Combine the results from Step 1 and Step 3. The intervals where both the conditions are satisfied, i.e. \(-3 < x < 3 \) and conditions in Step 3, are \([-2,1)\) and \([2,3)\).

Key Concepts

Natural Logarithm Constraints

In mathematics, the natural logarithm function, denoted as \( \ln \), has a unique behavior because it is only defined for positive arguments. This simply means that when we see \( \ln(x) \), the value inside the parentheses, \(x\), must be greater than zero.
In this problem, we have the natural logarithm applied to \( \operatorname{sgn}(9-x^2) \). The \( \operatorname{sgn} \) function (or sign function) will result in 1 when \(9-x^2 > 0\), which ensures that the argument of the logarithm is positive.

• The condition \(9-x^2 > 0\) simplifies to \(x^2 < 9\), which further means that \(x\) must lie within the interval \(-3 < x < 3\).

Understanding the constraints for the natural logarithm is integral to determining the valid domain of the function we are analyzing.

Square Root Constraints

The square root function, similarly to the logarithm, has its own domain restrictions. A square root must have a non-negative argument to be defined, as square roots of negative numbers are not real. In this exercise, the function is \( \sqrt{[x]^3 - 4[x]} \), where the term within the square root must be non-negative.

• The expression \([x]^3 - 4[x] \geq 0\) is resolved based on the integer values of \([x]\), the integral part function of \(x\).
• For each integer \([x]\) within the potential domain from the natural logarithm \((-3 < x < 3)\), we calculate \([x]^3 - 4[x]\) to ensure it is non-negative.

Depending on \([x]\), the evaluation results vary:

• \([x] = -2\), \([-1\), \(0\), and \(2\) give us non-negative values, which allows the function to be valid for these parts of the interval.
• On the contrary, \([x] = -3\) and \([x] = 1\) lead to negative results, indicating those must be excluded from the domain.

Integral Part Functions

The integral part function, denoted as \([x]\), represents the greatest integer less than or equal to \(x\). Understanding \([x]\) is essential for working through problems involving piecewise function behavior, like this one.

• For the domain \(-3 < x < 3\), the values of \([x]\) change at every integer point: from -2, -1, 0 up to 2.
• Each of these integers represents the integral part of \(x\) in its specific interval, which influences the expression within the square root \([x]^3 - 4[x]\).

By assessing \([x]\) at each segment, we determine when \([x]^3 - 4[x] \) is zero or positive, thereby refining the domain for which the original function is defined. Thus, mastering integral part functions is crucial for deciphering these segment-dependent constraints in functional domains.