Problem 9 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If the real valued function \(f(x)=\frac{a^{x}-1}{x^{n}\left(a^{x}+1\right)}\), is even then \(n\) equals (a) 2 (b) \(2 / 3\) (c) \(1 / 4\) (d) \(-1 / 3\)

Step-by-Step Solution

Answer

The correct answer is (a) 2.

Step 1: Understanding the Definition of an Even Function

A function is even if for every value of \(x\) in the domain of the function, the equation \(f(x) = f(-x)\) holds true. This is the starting point to determine the conditions that \(n\) must satisfy.

Step 2: Set Up the Equation for an Even Function

Given the function \(f(x) = \frac{a^x - 1}{x^n (a^x + 1)}\), we want \(f(x) = f(-x)\). This implies that \(\frac{a^x - 1}{x^n (a^x + 1)} = \frac{a^{-x} - 1}{(-x)^n (a^{-x} + 1)}\).

Step 3: Simplify the Expression for \(f(-x)\)

Substitute \(-x\) into the function: \(f(-x) = \frac{a^{-x} - 1}{(-x)^n (a^{-x} + 1)}\). Simplify \(a^{-x}\) as \(\frac{1}{a^x}\), which gives \(f(-x) = \frac{\frac{1}{a^x} - 1}{(-x)^n (\frac{1}{a^x} + 1)}\). This simplifies to \(\frac{1-a^x}{(-x)^n (1+a^x)}\).

Step 4: Decide When \(f(x) = f(-x)\)

Comparing \(f(x)\) and \(f(-x)\), we have \(\frac{a^x - 1}{x^n (a^x + 1)} = \frac{1-a^x}{(-x)^n (1+a^x)}\). Simplifying the equation: 1. The numerators are opposites: \((a^x - 1) = -(1-a^x)\). 2. The denominators become equal when \(x^n = (-x)^n\), which is true when \(n\) is even.The simplest even number is \(2\), as it is compatible with integer and standard polynomial expressions.

Key Concepts

Real Valued Functions

In mathematics, a real valued function is any function that takes real numbers as input and produces real numbers as output. These functions can be visualized easily on a two-dimensional plane, where the x-axis represents the input values and the y-axis represents the output values.

Real valued functions come in various forms and are foundational pieces in calculus and algebra. They can represent anything from simple linear equations to complex polynomial or trigonometric expressions.

• **Definition:** A real valued function is expressed as \( f: \mathbb{R} \to \mathbb{R} \), indicating that the domain and range include all real numbers.
• **Applications:** These functions are used in diverse fields such as physics, finance, and engineering to model behavior that changes continuously and predictably.
• **Notation:** Common notation includes \( f(x) = y \), where \( y \) is the function's output given the input \( x \).

This exercise involves a specified real valued function, allowing us to explore further into its properties, such as symmetry.

Symmetry in Functions

Symmetry in functions refers to the attribute where a function appears the same when looked at from different perspectives. Symmetry most commonly appears as either even or odd functions. Knowing if a function is symmetric can simplify analysis and problem solving.

Specifically, an even function satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain. This reflects mirror symmetry about the y-axis. For instance, consider the function \( f(x) = x^2 \). Replacing \( x \) with \(-x\), we find that \( (-x)^2 = x^2 \), verifying its even nature.

• **Even Functions:** Characterized by symmetry around the y-axis, meaning the graph looks identical on either side of this axis.
• **Odd Functions:** Where \( f(-x) = -f(x) \), they have rotational symmetry around the origin.
• **Graphical Features:** Analysis often involves checking graph symmetry or applying algebraic transformations to confirm evenness or oddness.

Understanding that our function could be even helped determine the value of \( n \), in ensuring symmetry.

Properties of Exponents

Exponents play a pivotal role in many mathematical functions, providing a compact notation to represent repeated multiplication. Mastering their properties can greatly simplify complex expressions and equations.

The function analyzed here contained terms like \( a^x \) and \( a^{-x} \), using exponentiation to describe its behavior. Understanding these properties aids in dissecting and simplifying such expressions. Key properties include:

• **Product of Powers:** \( a^{m+n} = a^m \cdot a^n \)
• **Quotient of Powers:** \( a^{m-n} = \frac{a^m}{a^n} \)
• **Power of a Power:** \( (a^m)^n = a^{mn} \)

For instance, in the context of \( a^{-x} \), it can be translated to \( \frac{1}{a^x} \) through the negative exponent rule. These principles helped simplify the equation to verify the symmetry condition \( f(x) = f(-x) \), ultimately solving for \( n = 2 \). This ensures that the exponents in both \( f(x) \) and \( f(-x) \) remain compatible, preserving the required even function property.