Problem 138 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The range of the function \(f(x)=\frac{\sin \left(\pi\left[x^{2}+1\right]\right)}{x^{4}+1}\), where \([\cdot]\) denotes the greatest integer function, is (A) \([0,1]\) (B) \([-1,1]\) (C) \(\\{0\\}\) (D) None of these

Step-by-Step Solution

Answer

(C) \{0\}.

Step 1: Understanding the Function Components

First, let's break down the function: \(f(x)=\frac{\sin \left(\pi\left[x^{2}+1\right]\right)}{x^{4}+1}\). The numerator involves the sine function of \(\pi[x^2+1]\), where \([\cdot]\) is the greatest integer function, or floor function, which rounds down to the nearest integer. The denominator is \(x^4 + 1\), which is always positive.

Step 2: Analyzing the Numerator

For the expression \(\sin(\pi [x^2+1])\), \([x^2 + 1]\) can only take integer values because it uses the greatest integer function. This means \(\sin(\pi n)\) for some integer \(n\). Since \(\sin(\pi n) = 0\) for any integer \(n\), the numerator becomes 0 for all \(x\).

Step 3: Determining the Denominator's Effect

The denominator \(x^4+1\) is always positive for all real numbers \(x\) because power 4 of any real number is non-negative, and adding 1 ensures it's positive. This means the division by \(x^4+1\) does not influence the zero value of the numerator.

Step 4: Concluding the Range

Since the numerator is zero, regardless of the value of \(x\), the entire function evaluates to zero. Therefore, the range of the function is \(\{0\}\), as it takes only one constant value.

Key Concepts

Greatest Integer Function

The greatest integer function, often represented as \([ \cdot ]\), is a fundamental mathematical concept. It is also called the floor function. This function takes any real number and rounds it down to the nearest integer. For instance, if you take 3.7, the greatest integer function \[ [3.7] = 3 \]. Similarly, for -2.4, we have \[ [-2.4] = -3 \].

• The key aspect of this function is its rounding down property.
• It transforms any number into the largest integer that is less than or equal to the original number.
• This property makes it crucial in analyzing functions like the one in our exercise.

In the problem, \[ [x^2 + 1] \] is used. No matter the value of \(x\), whether it's positive or negative, squaring it ensures the result is non-negative. Adding 1 raises the lowest possible integer outcome to 1 or more.
Because \(x^2 + 1\) is always non-integer (except for certain specific values), the greatest integer function changes it to the nearest lower integer. Understanding this helps us realize why \([x^2 + 1] = n\), and thus, \(\sin(\pi n)\) results in 0 for the numerator, maintaining a zero value throughout.

Numerator Analysis

The analysis of the numerator in functions is a vital step in understanding the function's overall behavior. In the exercise, the numerator is \(\sin(\pi [x^2 + 1])\).To break it down further:- The expression \([x^2 + 1]\) produces integer values due to the greatest integer function.- Applying \(\pi\) times an integer inside a sine function leads to interesting results.The sine function itself, \(\sin(\theta)\), returns a value between -1 and 1 for angles in radians.
However, the sine of any multiple of \(\pi\), such as \(\pi n\) where \(n\) is an integer, is precisely 0. This is because sine returns zero at every integer multiple of \(\pi\). Thus, regardless of \(x\), \(\sin(\pi [x^2 + 1])\) results in 0.With a zero in the numerator:

• The function itself simplifies significantly.
• Even when divided by any positive denominator, the function's output remains zero.

Denominator Properties

The denominator of a function is crucial since it can determine the overall function's domain and range. In this exercise, the denominator is \(x^4 + 1\). Evaluating its properties:

• For any real algebraic expression \(x^4\), squaring ensures the output is non-negative. Since squaring a number makes even negative inputs positive, \(x^4\) is always zero or greater.
• Adding 1 to \(x^4\) ensures that \(x^4 + 1\) is positive for all real values of \(x\).

The importance of these properties:- Because \(x^4 + 1\) can never be zero, the function \(f(x) = \frac{\sin(\pi [x^2 + 1])}{x^4+1}\) does not exhibit division by zero, avoiding undefined behavior.- It assures the safety of division and confirms that the denominator does not affect whether the function result in zero or not.Therefore, the consistent positiveness of \(x^4 + 1\) maintains the integrity of the division without complications.