Problem 129 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=\frac{9^{x}}{9^{x}+3}\). Then, (A) \(f(x)+f(1-x)=1\) (B) \(f(x)+f(1-x)=-1\) (C) \(f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)\) \(+f\left(\frac{3}{1996}\right)+\ldots+f\left(\frac{1995}{1996}\right)=998\) (D) \(f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)\) \(+f\left(\frac{3}{1996}\right)+\ldots+f\left(\frac{1995}{1996}\right)=997 \frac{1}{2}\)

Step-by-Step Solution

Answer

(A) is true; (D) is true.

Step 1: Understand the Function

The function given is \( f(x) = \frac{9^x}{9^x + 3} \). This function has the domain of all real numbers \( x \), and it is helpful to find the complement \( f(1-x) \) to solve the given choices.

Step 2: Compute f(1-x)

Substitute \( 1-x \) into the function: \( f(1-x) = \frac{9^{1-x}}{9^{1-x} + 3} \). This changes the denominator to \( 9 \cdot 9^{-x} + 3 \).

Step 3: Simplify f(x) + f(1-x)

Calculate \( f(x) + f(1-x) = \frac{9^x}{9^x + 3} + \frac{9 \cdot 9^{-x}}{9 \cdot 9^{-x} + 3} \). Simplify both terms by finding a common denominator: \((9^x + 3)(9^{1-x} + 3) = 9 + 3 \cdot 9^{1-x} + 9^x \cdot 3 + 9^x \cdot 9^{-x}\), which simplifies further to show it adds up to 1.

Step 4: Confirm f(x) + f(1-x) = 1

The expression simplifies to \( f(x) + f(1-x) = 1 \). Each term in the sum cancels out one from the other term when combined. Thus, option (A) \(f(x)+f(1-x)=1\) is correct.

Step 5: Examine the Options for Summation

Given \( f(x) + f(1-x) = 1 \), for the sequence \( x = \frac{k}{1996} \), each pair is complementary around \( \frac{1}{2} \). Therefore, \( f\left(\frac{k}{1996}\right) + f\left(\frac{1996-k}{1996}\right) = 1 \).

Step 6: Compute the Offered Sums

There are 1995 terms, but recognizing the complementary pairs leads to 997 full pairs of terms, each summing to 1. The sequence is from 1 to 1995, halving that gives 997, making the total \( 997 \). Remaining middle term at \( \frac{1}{2} \) would itself = \( \frac{1}{2} \).

Step 7: Validate Summation Totals

Option (D) states the correct amount: \( f\left(\frac{1}{1996}\right) + f\left(\frac{2}{1996}\right) + \ldots + f\left(\frac{1995}{1996}\right) = 997 \frac{1}{2} \). This includes the middle term at exactly \( \frac{1}{2} \).

Key Concepts

Function Properties

Functions are mathematical relations between two distinct sets where each element of the first set corresponds to exactly one element of the second set. Understanding function properties helps analyze and predict the behavior and relationship of different functions. For the given problem, the function we are working with is \( f(x) = \frac{9^x}{9^x + 3} \). This function involves an exponential expression, and its properties influence how it behaves over its domain, which consists of all real numbers.

• Domain and Range: The domain is all real numbers, as the format \( \frac{a}{a+b} \) ensures no division by zero. The values produced by the function are constrained by its components; as \( 9^x \) grows large, \( f(x) \) approaches a value of 1, and as it grows small, \( f(x) \) approaches 0.
• Symmetry and Complementarity: A notable property demonstrated is the complementarity, \( f(x) + f(1-x) = 1 \). This symmetry usually happens when a function and its reverse transform mirror each other around a central axis or point, providing a vital characteristic for further calculations.
• Exponential Components: With exponential expressions such as \( 9^x \) and \( 9^{1-x} \), the properties like one-to-one and onto behaviors could be affected by these growth factors.

Recognizing these properties is essential in solving complex problems by allowing easier manipulation and examination of functions.

Complementary Functions

Complementary functions are often used when two functions are intertwined in behavior or when particular properties interrelate them, as seen in this exercise. When functions satisfy an equation together, complementarity can simplify analysis or problem-solving.

• Understanding Complementary: Suppose two functions \( f(x) \) and \( g(x) \) are complementary if their sum or difference equals a constant. For instance, \( f(x) + f(1-x) = 1 \) is classic complementarity. It implies that each function behaves as a perfect counterpart to balance out the other.
• Using Complementary Functions in Problem Solving: When analyzing functions in complement, patterns and symmetries emerge. This can simplify the derivation process since you often only need to evaluate one half of a set due to the repeating nature of these functions.
• Role in Summation: Complementary functions are particularly effective in summation problems where paired terms from different ends of a range complement each other. In sum, such functions can significantly reduce computational complexity by pairing terms of sequences like \( f\left(\frac{k}{1996}\right) + f\left(1-\frac{k}{1996}\right) \).

By leveraging complementary properties, problem-solving becomes more straightforward and more efficient, often reducing what would be an overwhelming set of calculations into a manageable task.

Summation of Series

In mathematics, summation of series is the process of adding up all elements from a sequence of numbers. Different techniques simplify these tasks significantly, especially when dealing with functions and sequences.

• Understanding Series and Sequences: In the context of this exercise, we're interested in specific values of the function evaluated at fractional inputs. Such series often have underlying patterns and structures, such as symmetry, that can be utilized to reduce computation.
• Complementary Pairing in Series: Using complementary function properties, as seen with \( f\left(\frac{k}{1996}\right) + f\left(1-\frac{k}{1996}\right) \), reduces the terms effectively by pairing to make regularly summing results, in this case, 1, per pair.
• Application: Evaluate how the series behaves over its range, recognizing sums can be simplified here due to a tightly-knit complementary pattern and balance that emerged, leading to integer values with minimal effort due to property benefit, such as exactly 997, plus a middle-term adjustment.

Understanding these concepts is crucial when dealing with combinations of functions and sequences, offering insight into efficient problem-solving and analysis.