Problem 31 - 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, [ ] 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 \(\{0\}\).

Step 1: Understand the Function

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

Step 2: Analyze the Greatest Integer Function

For any real number \(x\), \(x^2\) is non-negative. Thus, \(x^2 + 1\) ranges from \(1\) to \(\infty\). The greatest integer function \([x^2 + 1]\) will then be an integer \(n\) greater than or equal to 1.

Step 3: Evaluate the Sine Function

For integer \(n\), \(\sin(\pi n)\) equals 0 because the sine of any multiple of \(\pi\) is 0. Therefore, \(\sin(\pi[x^2 + 1]) = 0\) for any \(x\).

Step 4: Express the Function

Since the sine function in the numerator is 0 for any \(x\), we can see that \(f(x) = \frac{0}{x^4 + 1} = 0\) for all \(x\).

Step 5: Determine the Range

The range of a function is the set of possible values for \(f(x)\) as \(x\) varies over all real numbers. Since \(f(x) = 0\) for all \(x\), the range of this function is \(\{0\}\).

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted by the square bracket symbols \([x]\), is a key mathematical concept that helps in stepping down a real number to its largest previous integer. In a more formal sense, for a given real number \(x\), the expression \[ [x] = n \] gives the greatest integer \(n\) such that \(n \leq x\).
This function has unique properties:

• The output is always an integer. So, this function essentially "chops off" the decimal part and retains only the integer portion.
• It is non-decreasing as the \(x\) increases. That means \([x] < [x+1]\) if \(x\) is not already an integer.
• This function is often used when discrete values are necessary from continuous inputs.

In the exercise, \([x^2 + 1]\) ensures that we always have an integer \(n\) starting from \(1\). This is crucial for functions involving integers, such as sine, helping determine fixed values, like in our example where \(\sin(\pi n) = 0\) for all integer \(n\).

Trigonometric Functions

Trigonometric functions relate angles of triangles to ratios between sides. The sine function, one of the key trigonometric functions, oscillates between \(-1\) and \(1\) and repeats its pattern every \(2\pi\) radians. An essential property is that sine of multiples of \(\pi\), is zero. This plays a critical role in the given problem.
When we observe the function \(f(x) = \frac{\sin(\pi[x^2 + 1])}{x^4 + 1}\), the sine component \(\sin(\pi[x^2 + 1])\) is evaluated at integer multiples of \(\pi\), making it zero. This results because:

• For any integer \(n\), \[ \sin(\pi n) = 0 \].
• The sine function's behavior simplifies the analysis of the numerator, leading the entire function to become zero.
• This effect is identical and amplified when combined with other functions like the greatest integer function in our exercise.

Understanding these behaviors of the sine function is critical in determining the range for any combined function, allowing us to predict its effects over different inputs.

Function Analysis

Function analysis involves examining the various characteristics of a function, such as domain, range, limits, and behavior. In general, it involves thoroughly understanding how a function behaves over its inputs to predict its outputs. Breaking down each feature helps unravel complex functions into understandable parts.
In the function \(f(x) = \frac{\sin(\pi[x^2 + 1])}{x^4 + 1}\), function analysis is executed by:

• Determining the domain: The function is defined for all real numbers \(x\) because both the sine and polynomial in the denominator are defined everywhere.
• Finding its behavior: Since \(\sin(\pi[x^2 + 1]) = 0\) for all \(x\), the behavior of the function simplifies to being zero regardless of \(x\).
• Identifying the range: Since every output of \(f(x)\) is zero, the function's range is simply \({0}\).
• Ensuring the denominator isn't zero which \(x^4 + 1\) fulfills as it's always positive.

Function analysis provides clarity about what a function can output given all possible inputs, a basic yet critical skill in mathematics.