Problem 13 - Calculus for JEE ADVANCED

Question

\(f(x) \frac{2^{x}+1}{4^{x}-1}=\frac{1}{2^{x}-1}\) is one-one function.

Step-by-Step Solution

Answer

Yes, \( f(x) \) is a one-one function as it satisfies the injective condition.

Step 1: Understand the One-One Function Definition

A function is one-one, or injective, if each element of the domain maps to a unique element of the range. This means that if \( f(a) = f(b) \), then \( a = b \) must hold true.

Step 2: Equate and Simplify

Given the function equation \( f(x) = \frac{2^x + 1}{4^x - 1} = \frac{1}{2^x - 1} \), simplify and analyze its structure. Start with expressing \( 4^x \) as \((2^x)^2\), leading to \( f(x) = \frac{2^x + 1}{(2^x)^2 - 1} \). The denominator factorizes as \((2^x - 1)(2^x + 1)\), thus \( f(x) = \frac{1}{2^x - 1} \), confirming the given expression.

Step 3: Verify Injectivity

To verify that \( f(x) \) is one-one, assume \( f(a) = f(b) \) leading to \( \frac{1}{2^a - 1} = \frac{1}{2^b - 1} \). Simplifying gives \( 2^a - 1 = 2^b - 1 \), which implies \( 2^a = 2^b \). Since \( 2^x \) is an exponential function, this results in \( a = b \), affirming injectivity.

Key Concepts

Exponential Functions

An exponential function is a type of mathematical function in which an independent variable, often denoted as \( x \), appears in the exponent. This means the function grows or decays at a rate proportional to its current value. A classic example is \( 2^x \), where the base is 2 and the exponent is the variable \( x \). The defining characteristic of exponential functions is their rapid increase or decrease, resulting in distinct growth patterns compared to linear or polynomial functions.
Exponential functions have several properties:

• They are continuous and smooth, meaning they have no breaks, holes, or jumps.
• The base of the exponential (like 2 in \( 2^x \)) must be a positive number other than 1.
• They are always positive for all real numbers \( x \).

Understanding and working with exponential functions is crucial in various fields like finance for compound interest, biology for population growth, and physics for radioactive decay.

Factorization

Factorization involves breaking down complex expressions into simpler, multipliable numbers or expressions that, when multiplied together, give the original term. This mathematical process can simplify equations and is often employed to solve calculus problems by reducing the complexity of expressions.
Consider the expression \( 4^x \) which we factorized as \((2^x)^2\). This step is crucial as it reveals more about the structure of the function by identifying it as the product \((2^x - 1)(2^x + 1)\). These factors then help in simplifying the function and examining its properties.

• Factorization allows us to simplify complex mathematical expressions.
• It helps in finding solutions to equations, especially polynomial ones.
• In injective function determination, it aids in reducing expressions to a form where injectivity can be easily assessed.

Effectively utilizing factorization is key to understanding and analyzing complex mathematical expressions and their properties.

Function Injectivity

A function is called injective, or one-one, when each element of the domain maps to a distinct and unique element in the range. This important property ensures that no two different inputs yield the same output in the function. To check if a function like \( f(x) = \frac{1}{2^x - 1} \) is injective, we assume \( f(a) = f(b) \) and aim to show that this implies \( a = b \).
Here’s a closer look at injective functions:

• Injective functions help in establishing a clear relationship between inputs and outputs without ambiguity.
• One method to prove injectivity involves setting \( f(a) = f(b) \) and simplifying to check if this results in \( a = b \).
• Injective properties are vital in various branches of mathematics, including algebra and calculus.

Understanding injectivity helps in grasping the fundamental nature of mappings and transformations within mathematical frameworks.

Simplification Techniques

Simplification harbors methods that reduce mathematical expressions to more manageable forms without changing their value. This process is crucial in solving equations, proving properties like injectivity, and making complex calculations easier to digest.
One technique involves observing expressions for potential factorization opportunities, as used in simplifying \( 4^x - 1 \) to \((2^x - 1)(2^x + 1)\). Simplification might also include canceling common factors, rearranging terms, or substituting equivalent expressions.

• Simplification can involve factorization, like turning complex polynomial expressions into factorized forms.
• Cancelling out common terms simplifies rational functions by eliminating complexity.
• Using substitution can replace cumbersome expressions with simpler alternatives.

Mastering simplification techniques is crucial for easing otherwise daunting mathematical challenges, helping you find solutions more efficiently and accurately.