Problem 52 - A Complete Resource Book in Mathematics for JEE Main

Question

If \([x]\) denotes the integral part of \(x\), then the domain of the function \(f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left[\log _{12}\left(x^{2}-5 x+5\right)\right]\) is (a) \(\left(-\sqrt{\frac{5}{2}},-1\right]\) (B) \(\left[1, \sqrt{\frac{5}{2}}\right)\) (C) \(\left(-\sqrt{\frac{5}{2}},-1\right] \cup\left[1, \sqrt{\frac{5}{2}}\right)\) (D) None of these

Step-by-Step Solution

Answer

Correct answer is (C): \((-\sqrt{\frac{5}{2}}, -1) \cup [1, \sqrt{\frac{5}{2}})\).

Step 1: Understanding the function components

The function is composed of two main parts: \( \sin^{-1}([2x^2 - 3]) \) and \( \log_2([\log_{12}(x^2 - 5x + 5)]) \). To define the domain, we must ensure both expressions are defined and real.

Step 2: Conditions for sin^{-1}

The expression \( \sin^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we need \([2x^2 - 3] \leq 1\) and \([2x^2 - 3] \geq -1\). This means \( -1 \leq 2x^2 - 3 < 2 \).

Step 3: Solve for x in sin^{-1} condition

Solving \(-1 \leq 2x^2 - 3 < 2\), we get: \(2 \leq 2x^2 < 5\). Dividing by 2, this gives \(1 \leq x^2 < 2.5\). Thus, \(- \sqrt{2.5} \leq x \leq -1\) or \(1 \leq x < \sqrt{2.5}\).

Step 4: Conditions for log expression

The expression \( \log_2(z) \) is defined for \( z > 0 \). This requires \([\log_{12}(x^2 - 5x + 5)] > 0\). Since the integral part \([w] > 0\), \(w > 1\), meaning \(x^2 - 5x + 5 > 12\).

Step 5: Solve for x in log condition

Solving the inequality \(x^2 - 5x + 5 > 12\), we rearrange to \(x^2 - 5x - 7 > 0\). Solving for \(x\), the roots are approximately \(-1.29\) and \(6.29\), meaning \(x < -1.29\) or \(x > 6.29\).

Step 6: Combining both conditions for domain

The overlap of valid solutions for both parts of the function is the domain. From earlier steps, we know it must satisfy both \(- \sqrt{2.5} \leq x < -1.29 \) and \(1 < x < \sqrt{2.5}\). The combined domain is: \((- \sqrt{2.5}, -1.29) \cup (1, \sqrt{2.5})\).

Key Concepts

Integral Part

The integral part of a number, often symbolized \([x]\), is the integer part of a real number \(x\). It's effectively the whole number value that is less than or equal to \(x\). This is commonly referred to as the floor function, denoted by \( \lfloor x \rfloor \). For example:\

• \([2.7] = 2\), because 2 is the largest integer less than or equal to 2.7.
• \([-3.4] = -4\) because -4 is the largest integer less than or equal to -3.4.

In the context of our function \[f(x)=\sin^{-1}[2x^2-3]+\log_2[\log_{12}(x^2-5x+5)]\], the integral part ensures that the values of \(2x^2 - 3\) and \(\log_{12}(x^2 - 5x + 5)\) are integers. This is crucial because \(\sin^{-1}([2x^2-3])\) requires the result of \([2x^2-3]\) to be within the range \([-1, 1]\) to be valid. Additionally, \(\log_2([\log_{12}(x^2 - 5x + 5)])\) necessitates that \([\log_{12}(x^2 - 5x + 5)]\) is greater than zero since the logarithm of zero or a negative number is undefined.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the trigonometric functions such as sine, cosine, and tangent. They are used to determine angles from trigonometric ratios. The specific inverse function in our context is the inverse sine, denoted as \( \sin^{-1}(x) \) or arcsin. This function is especially pertinent in calculating an angle when its sine value is known. The range of \( \sin^{-1}(x) \) is \([-\pi/2, \pi/2]\) and it accepts inputs from the set\([-1, 1]\).
For instance, if \( \sin(\theta) = 0.5 \), then \( \theta = \sin^{-1}(0.5) \), where \(\theta\) falls between \(-\pi/2\) and \(\pi/2\).
Now, when using the inverse sine with the integral part \([2x^2 - 3]\), the integral part restricts \(2x^2 - 3\) to integer values. Thus, the value \([2x^2 - 3]\) must be within \([-1, 1]\), because \( \sin^{-1}(x) \) can only accept values in this range. This constraint is essential for defining the domain of the function we are considering.

Logarithmic Inequalities

Logarithmic inequalities involve comparing logarithmic expressions. These inequalities focus on ensuring the arguments of the logarithm are permissible, which primarily means they must be positive. In our function, \( \log_2([\log_{12}(x^2 - 5x + 5)]) \), it is necessary for the inner logarithmic expression \([\log_{12}(x^2 - 5x + 5)]\) to be greater than zero, ensuring that the expression under the second logarithm is positive.
The nested inequality \( [\log_{12}(x^2 - 5x + 5)] > 0 \) implies \( \log_{12}(x^2 - 5x + 5) > 1 \), meaning \(x^2 - 5x + 5\) must be greater than 12, which translates to solving \(x^2 - 5x - 7 > 0\). This is critical because a positive logarithmic result ensures the validity and realness of the logarithm. By analyzing and solving these inequalities, we determine the regions where the combined expressions defining \(f(x)\)'s domain are valid.