Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(x)=\max .\\{(1-x),(1+x), 2\\}, \forall x \in \mathrm{R}\). Then (A) \(f(x)=\left\\{\begin{array}{lc}1+x, & x \leq-1 \\ 2, & -1

Step-by-Step Solution

Answer

The correct function is option (C).

Step 1: Understanding the function components

Given the function is defined as \( f(x) = \max \{ (1-x), (1+x), 2 \} \). This means for any value of \( x \), \( f(x) \) will take the greatest value among \( (1-x) \), \( (1+x) \), and \( 2 \). We aim to identify which piecewise definition matches this condition.

Step 2: Evaluating function behavior for different intervals

To determine how the function behaves, examine each component over intervals that naturally emerge:1. **Interval \( x \leq -1 \):** - \( 1-x \) becomes greater as \( x \) decreases. - \( 1+x \) becomes less significant since it remains negative. - Here, typically \( 1-x \) stays larger than \( 2 \). So, \( f(x) = 1+x \) does not apply; instead, for positive \( 1-x \) exceeds both.2. **Interval \( -1 < x < 1 \):** - Within this interval, \( 2 \) is always greater than both \( (1-x) \) and \( (1+x) \). - Therefore, \( f(x) = 2 \). 3. **Interval \( x \geq 1 \):** - \( 1+x \) increases and thus dominates over \( (1-x) \) and \( 2 \). - Consequently, \( f(x) = 1+x \).

Step 3: Constructing the piecewise function

Based on our analysis:- For \( x \leq -1 \): \( f(x) = 1-x \).- For \( -1 < x < 1 \): \( f(x) = 2 \).- For \( x \geq 1 \): \( f(x) = 1+x \).This corresponds to option (C).

Key Concepts

maximum function

When dealing with a maximum function, you're working with an operation that selects the highest value from a set of numbers. In the given exercise, the function is expressed as \( f(x) = \max \{ (1-x), (1+x), 2 \} \). This means that for every input \( x \), \( f(x) \) will be the greatest of \((1-x)\), \((1+x)\), and \(2\).

Here's how it works: each expression, \((1-x), (1+x), \) and \(2\), represents a separate line or parabola on a graph. At any point \( x \), the maximum function takes the highest value among these expressions. This is visually like looking at the top edge of a set of curves graphed together. The "top" in this context is whichever expression outputs the highest number for that specific \( x \).

Understanding maximum function helps you manage and predict behavior in complex piecewise functions. It's crucial when splitting a function into different relevant pieces, based on which term is the largest.

interval analysis

Interval analysis is the process of studying the behavior of a function within specified ranges of the independent variable—in this case, \( x \). By breaking down the broader domain into intervals, you can understand the function's behavior more accurately.

In the exercise, the function \( f(x) \) is analyzed over three main intervals:

• **Interval \( x \leq -1 \)**: Here, \( (1-x) \) emerges as the dominant term since it decreases less rapidly than \(1+x\), thus staying greater than both the second term and 2.
• **Interval \( -1 < x < 1 \)**: Within this window, the constant value \(2\) is higher than both linear terms \((1-x)\) and \((1+x)\).


• **Interval \( x \geq 1 \)**: The term \( (1+x) \) becomes significant because it increases linearly, surpassing \((1-x)\) and the constant value 2.

Conducting interval analysis allows you to construct the piecewise function accurately, identifying which expression dominates in each specific interval.

function behavior

Understanding the behavior of a piecewise function means knowing how its output changes according to different inputs. It's about observing how the function behaves across its domain and in its different regions.

Consider the function \( f(x) = \max \{ (1-x), (1+x), 2 \} \). Its behavior varies as follows:

• For values \( x \leq -1 \), the function prefers \((1-x)\), because as \(x\) decreases, \((1-x)\) increases.
• For the interval \(-1 < x < 1\), the function is constant, steadily outputting 2, as this is the peak value within this range.
• For \( x \geq 1 \), the term \( (1+x) \) drives the function's behavior, increasing as \(x\) grows larger.

Recognizing these distinct behaviors is fundamental. It tells you whether you're dealing with an increasing, decreasing, or constant segment of the function. This insight is essential for sketching the graph accurately and comprehending the overall picture of how the function operates across its full scope.