Problem 103 - A Complete Resource Book in Mathematics for JEE Main

Question

The domain of the function \(f(x)=[\sin x] \cos \left(\frac{\pi}{[x-1]}\right)\) (A) \((1,2)\) (B) \(R-[1,2)\) (C) \(R-(1,2)\) (D) None of these

Step-by-Step Solution

Answer

The domain is choice (C) \(R - (1,2)\).

Step 1: Determine domain constraints for \([\sin x]\)

The expression \([\sin x]\) gives the integer part (floor function) of \(\sin x\). Since \(\sin x\) ranges from -1 to 1 for all real \(x\), the integer part can be -1, 0, or 1. There are no additional restrictions on \(x\) from this term.

Step 2: Analyze domain constraints for \(\frac{\pi}{[x-1]}\)

The expression \([x-1]\) represents the integer part (floor function) of \(x-1\). For the function \(\cos\left(\frac{\pi}{[x-1]}\right)\) to be defined, \([x-1]\) must not be zero to avoid division by zero. Therefore, \(x-1\) should not be an integer where its floor part results in zero. This happens when \(x-1 \in [0,1)\), hence \(x \in [1,2)\). We need to exclude \(x \in [1,2)\) from the domain.

Step 3: Combine derived constraints to find the domain

From Step 1, the integer part of \(\sin x\) doesn't restrict \(x\) further. From Step 2, we determined that \(x \in [1,2)\) must be excluded to avoid division by zero in the cosine function. Thus, the domain of the function is all real numbers except the interval \((1,2)\).

Step 4: Identify correct answer choice

Comparing the derived domain \(R - (1,2)\) with the provided answer choices, option (C) \(R - (1,2)\) correctly represents the domain where the function is defined.

Key Concepts

floor function

The floor function, often denoted as \([x]\), is a mathematical function that takes a real number and rounds it down to the nearest integer less than or equal to that number. This feature plays a significant role when analyzing the domain of the function \(f(x)=[\sin x] \cos \left(\frac{\pi}{[x-1]}\right)\). In the context of \([\sin x]\), because \(\sin x\) ranges between -1 and 1, the floor function could result in three possible values: -1, 0, or 1. These values correspond to the whole numbers that lie within the range of the sine function, which continue to have crucial implications when determining the domain of functions involving trigonometric elements. Because the floor function simply maps to these integers without any further restriction, this component of the function does not introduce additional domain constraints.

trigonometric functions

Trigonometric functions, such as sine and cosine, are periodic functions fundamental in mathematics. In \(f(x)=[\sin x] \cos \left(\frac{\pi}{[x-1]}\right)\), the sine and cosine functions are combined with a floor function and a quotient. The sine function, represented by \(\sin x\), oscillates between -1 and 1, influencing the floor function as discussed earlier. Meanwhile, the cosine term, \(\cos\left(\frac{\pi}{[x-1]}\right)\), is influenced directly by the floor function applied to \(x-1\). The cosine function is periodic and defined for all real arguments, except when the argument causes undefined behavior, such as division by zero. This is why it is crucial to consider the behavior of \([x-1]\); when \([x-1]\) is zero, the denominator becomes zero, leading to undefined results. Thus, avoiding values of \(x\) where \([x-1]\) is zero is imperative in determining the correct domain for the function.

domain constraints

Function domain refers to the set of all possible input values (\(x\)-values) for which the function is defined. When determining the domain of \(f(x)=[\sin x] \cos\left(\frac{\pi}{[x-1]}\right)\), several constraints come into play:

• The sine component \([\sin x]\) allows all real \(x\) values since it only rounds the value of sine to a nearest lower whole number, such as -1, 0, or 1.
• The more critical constraint arises from the cosine portion \(\cos\left(\frac{\pi}{[x-1]}\right)\). Here, division by zero would occur if \([x-1]=0\), which happens if \(x\) is an integer within the interval \([1,2)\).

Thus, the function is undefined for values of \(x\) in \([1,2)\), meaning \(x\) should not lie in this range. By excluding these values, we derive the domain: all real numbers except \([1,2)\). This ensures no division by zero occurs within the cosine function, providing a valid and comprehensive domain for \(f(x)\). By understanding each component and its effect, you can accurately determine the overall domain of a complex function.