Problem 31 - 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, [ ] is greatest integer function, is (A) \([0,1]\) (B) \([-1,1]\) (C) \(\\{0\\}\) (D) None of these

Step-by-Step Solution

Answer

The range of the function is \(\{0\}\), so the answer is (C).

Step 1: Understanding the Function Components

The function given is \( f(x) = \frac{\sin\left(\pi\left[ x^2 + 1 \right]\right)}{x^4 + 1} \), where \([x^2 + 1]\) denotes the greatest integer function. This function evaluates to the greatest integer less than or equal to \(x^2 + 1\).

Step 2: Determine Range of the Sine Term

The term \(\sin(\pi\left[ x^2 + 1 \right])\) depends on the value of \(\left[ x^2 + 1 \right]\). The value of \(\left[ x^2 + 1 \right]\) is an integer, so \(\sin(n\pi)\) for any integer \(n\) is \(0\). Therefore, \(\sin(\pi \left[ x^2 + 1 \right]) = 0\).

Step 3: Analyzing the Denominator

The denominator \(x^4 + 1\) is a polynomial expression which is always positive for all real \(x\) due to the nature of even powered terms and the addition of 1.

Step 4: Evaluate the Range of the Function

As we determined, the numerator is constantly \(0\) due to the periodic and Integer property of sine of multiples of \(\pi\), the entire function \(f(x) = \frac{0}{x^4 + 1} = 0\).

Step 5: Determine the Correct Answer

Given that the function outputs \(0\) for all real \(x\), the range of the function is \(\{0\}\). Thus, option (C) is the correct answer.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([ \cdot ]\), is a mathematical concept that assigns to each real number the greatest integer less than or equal to that number. It is also known as the floor function. The behavior of this function is somewhat similar to rounding down to the nearest whole number.

• For example, for the input 3.7, the greatest integer function outputs 3.
• For an input of -2.5, the result would be -3.

In the context of the given function, \([x^2 + 1]\), this means we first calculate the value of \(x^2 + 1\), and then take the greatest integer less than or equal to this result. Since the value of \(x^2 + 1\) is always at least 1 for any real \(x\), \([x^2 + 1]\) yields an integer starting from 1 upwards depending on \(x\). Understanding how this impacts functions like \(f(x)\) is vital, especially for determining behavior linked to sine functions over integer multiples of \(\pi\).

Sine Function

The sine function is a fundamental trigonometric function that takes an angle as input and outputs a value between -1 and 1. This is due to the nature of sine waves which oscillate between these two values as they make a complete cycle from 0 to 360 degrees or 0 to \(2\pi\) radians.
An important property of the sine function is its periodicity. It completes a cycle by repeating its pattern after every \(2\pi\) radians. When considering integer multiples of \(\pi\), such as in our original exercise, things become quite specific:

• At each integer multiple of \(\pi\), the sine function evaluates to 0.

Thus for the sine term in \(f(x) = \frac{\sin(\pi[x^2+1])}{x^4+1}\), since \([x^2+1]\) results in an integer, \(\sin(\pi \times \text{integer}) = 0\), simplifying our problem immensely. This key property is crucial, as it determines that the output of the sine portion, and thus the overall function, remains zero across all \(x\).

Polynomial Expression

A polynomial expression involves sums and differences of powers of a variable, each multiplied by a coefficient. The expression gets its name from the multiple (poly) terms (nomials) in it. The polynomial in the given function is in the denominator: \(x^4 + 1\).

• Any expression of the form \(x^n\) (where \(n\) is a non-negative integer) is a polynomial.
• For \(x^4 + 1\), it is important to note it is always positive for any real \(x\). This is due to even powers returning non-negative results, and the addition of 1 ensures it never diminishes to zero or a negative number.

This characteristic means the denominator in \(f(x) = \frac{\sin(\pi [x^2 + 1])}{x^4 + 1}\) is never zero, which is significant in avoiding undefined behavior. Consequently, throughout all \(x\), the polynomial keeps the function well-defined and contributes to the constant 0 output that arises from the zeroed numerator.