Problem 75 - Calculus for JEE ADVANCED

Question

For odd function, $$ \begin{aligned} f(x) &=-f(-x) \\ &=- \begin{cases}\sin (-x)+\cos (-x) & 0<-x<\pi / 2 \\ a, & \quad-x=\pi / 2 \\ \tan ^{2}(-x)+\operatorname{cosec}(-x), & \pi / 2<-x<\pi\end{cases} \\ &= \begin{cases}\sin x-\cos x, & -\pi / 2

Step-by-Step Solution

Answer

The odd function is expressed as \(f(x) = sin(x) - cos(x)\), \(f(x) = -a\), or \(f(x) = -tan^2(x) + cosec(x)\) depending on \(x\)'s value.

Step 1: Understanding Odd Functions

An odd function satisfies the property that \( f(x) = -f(-x) \). Our goal is to express \( f(x) \) for the domain of negative \( x \) using the given piecewise function for the positive \( x \) values.

Step 2: Rewrite the Piecewise Function for Negative Inputs

Given the function for \( 0 < -x < \pi/2 \), the expression inside becomes \( \ sin(-x) + cos(-x) \). Using odd and even properties, rewrite this as \( -sin(x) + cos(x) \) which simplifies to \( sin(x) - cos(x) \) for \( -\pi/2 < x < 0 \).

Step 3: Determine Value at Specific Points

For \( -x = \pi/2 \), we have \( x = -\pi/2 \). So, \( f(-\pi/2) = -a \).

Step 4: Rewrite for Larger Negative Intervals

For the interval \( \pi/2 < -x < \pi \), rewrite \( tan^2(-x) + cosec(-x) \). This uses \( tan(x) = -tan(-x) \) and \( cosec(x) = -cosec(-x) \), becoming \( -tan^2(x) + cosec(x) \) for \( -\pi < x < -\pi/2 \).

Step 5: Summarize the Piecewise Function

For the domain \(-\pi < x < 0\), we have:- \(sin(x) - cos(x)\) for \(-\pi/2 < x < 0\).- \(-a\) for \(x = -\pi/2\).- \(-tan^2(x) + cosec(x)\) for \(-\pi < x < -\pi/2\).

Key Concepts

Piecewise Functions

Piecewise functions are a fascinating concept in mathematics where a function is defined by different expressions depending on the interval of the input value. Think of it as a function with multiple "pieces". This concept is especially useful for dealing with complex functions that might not have a simple single expression to describe them across their entire domain.
In our given exercise, the piecewise function is defined by different expressions in separate intervals, specifically for positive and negative values of \(x\). For example:

• For \(0 < -x < \pi/2\), the function expression is \(\sin(-x) + \cos(-x)\).
• For \(-\pi/2 < x < 0\), it transforms to \(\sin(x) - \cos(x)\).
• A different approach is used for \(\pi/2 < -x < \pi\) and its negative counterpart.

Understanding how to piece these parts together is crucial for analyzing how the function behaves across different regions.

Trigonometric Identities

When dealing with trigonometric functions, understanding key identities is crucial. These identities allow us to transform and simplify expressions, making them essential tools in this exercise.
Some important identities used include:

• Odd functions: Functions like \(\sin(x)\) which satisfy \(\sin(-x) = -\sin(x)\).
• Even functions: Functions like \(\cos(x)\) for which \(\cos(-x) = \cos(x)\).
• Tangent and cosecant properties: Understanding that \(\tan(-x) = -\tan(x)\) and \(\cosec(-x) = -\cosec(x)\) helps rewrite expressions efficiently.

By using these identities, we smoothly transition from positive to negative intervals in the function's domain, such as when \(\sin(-x) + \cos(-x)\) becomes \(\sin(x) - \cos(x)\) for \(-\pi/2 < x < 0\). These transformations are necessary to express the function for negative \(x\) and heed the properties of odd functions.

Function Transformation

Transformation in the context of functions refers to shifting, reflecting, stretching or compressing a graph. In our problem, the transformation process is visible when transitioning between positive and negative interval expressions of the piecewise function.
One specific transformation involved in the exercise is:

• Reflection: Based on the property of odd functions \(f(x) = -f(-x)\), reflecting across the origin is a key step. This is seen when \(\sin(-x) + \cos(-x)\) turns into \(\sin(x) - \cos(x)\).

Understanding these transformations allows for clear visualization and manipulation of functions to adhere to mathematical properties like those of odd functions. This way, functions maintain their true nature, even when expressed uniquely in different domains.