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

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

The range is \(\{0\}\) (Option C).

Step 1: Understand the Function Components

The function given is \(f(x) = \frac{\sin ( \pi [x^2 + 1])}{x^4 + 1} \). It consists of two main parts: the sine function in the numerator \(\sin(\pi [x^2 + 1])\), and the polynomial in the denominator \(x^4 + 1\). The greatest integer function \([\cdot]\) modifies the input to the sine function.

Step 2: Analyze the Greatest Integer Function

The term \([x^2 + 1]\) indicates the greatest integer less than or equal to \(x^2 + 1\). Since \(x^2\) is always non-negative, \([x^2 + 1]\) starts from 1 upwards. For any real \(x\), \(x^2\) can increase indefinitely, leading to \([x^2 + 1]\) being any integer 1, 2, 3, etc.

Step 3: Determine the Values of the Sine Function

The sine function \(\sin ( \pi k )\) only depends on integer values \(k\). Notably, for integer \(k\), \(\sin (\pi k)\) equals 0, because \(\sin(\pi), \sin(2\pi), \sin(3\pi),...\) are all 0.

Step 4: Examine the Denominator for Domain Issues

The expression \(x^4 + 1\) is always positive (since \(x^4 \geq 0\)), meaning the function is defined for all real \(x\). There are no restrictions in the domain based on the denominator going to zero.

Step 5: Identify the Range of the Function

Since \(\sin(\pi [x^2 + 1])\) is always 0 as long as \([x^2 + 1]\) is an integer, the function \(f(x)\) becomes \(\frac{0}{x^4 + 1} = 0\). Regardless of what the denominator evaluates to (always positive and finite), the entire function evaluates to zero.

Key Concepts

Greatest integer function

The greatest integer function, denoted by \([\cdot]\), is a function that rounds down to the nearest integer. Essentially, it takes any real number and finds the greatest integer either less than or equal to it.
For example:

• For a number like 3.7, the greatest integer function will return 3.
• For -1.2, it will return -2.

This function is significant because it can simplify the evaluation of other functions by converting continuous values into discrete steps. In the context of the exercise, \([x^2 + 1]\) adjusts \((x^2 + 1)\), a continuous value, into an integer. The value of \([x^2]\) starts at 0 (since \([x^2] \geq 0\) for all real numbers) and goes up as \([x^2]\) increases.
This effect means that as \([x^2 + 1]\) processes the input, it always evaluates to integers such as 1, 2, 3, etc. It determines the input to the sine function in the problem, significantly impacting the final value of the function.

Trigonometric functions

Trigonometric functions are kingpins when it comes to modeling periodic phenomena. In mathematics, these include \(\sin, \cos,\) and \(\tan\), among others.
Our exercise focuses on the sine function, \(\sin(\theta)\). The sine of any angle \(\theta\) oscillates between -1 and 1. However, when the input to the sine function is an integer multiple of \(\pi\), such as \(\pi k\) where \(k\) is an integer, the sine function takes on specific values.

• When \(k\) is 0, 1, 2, 3, and so on, \(\sin(\pi k)\) is always 0.

This happens because these points correspond to the origin and its repetitions on the unit circle, leading to a value of 0. In the context of the exercise, this characteristic means the numerator in the function always equals zero, simplifying the evaluation of \(f(x)\).

Polynomial functions

Polynomial functions are expressions involving sums of powers of variables. Common examples include quadratic (degree 2), cubic (degree 3), and quartic (degree 4) polynomials.
In our problem, the denominator \(x^4 + 1\) exemplifies a quartic polynomial function:

• It holds that \(x^4\) is non-negative for all real numbers \(x\), given that any number raised to an even power is positive.
• Moreover, adding 1 ensures \(x^4 + 1\) always remains positive for all \(x\).

This property guarantees the polynomial never reaches zero, ensuring that the denominator does not introduce any domain restrictions or undefined points in the function. In other words, no matter the value of \(x\), \(x^4 + 1\) does not disrupt the calculation of \(f(x)\), which is why the function is defined for all real numbers.