Problem 107 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(\begin{aligned} f(x) &=-1, \quad-2 \leq x \leq 0 \\ &=x-1, \quad 0

Step-by-Step Solution

Answer

The function \(g(x)\) is given by: \(\begin{aligned} g(x) &= |x|, \quad -2 \leq x \leq 0 \\\ &= 2|x - 1|, \quad 0 < x \leq 2 .\end{aligned}\) Its domain is \(-2 \leq x \leq 2\) and its range is \(0 \leq g(x) \leq 2\). The function \(\phi(x)\) is given by: \(\begin{aligned} \phi(x) &= 2||x| - 1| + |x|, \quad -2 \leq x \leq 0 \\\ &= 2||x| - 1| + 2|x - 1|, \quad 0 < x \leq 2 .\end{aligned}\) Its domain is \(-2 \leq x \leq 2\) and its range is \(0 \leq \phi(x) \leq 4\).

Step 1: Determine the function \(g(x)\)

To determine the function \(g(x)\), we have to find \(f(|x|)\) and \(|f(x)|\), and then add them together. We'll solve this separately for the two intervals where the piecewise function \(f(x)\) is defined. 1. When \(-2 \leq x \leq 0\): We have \(f(x) = -1\), so \(|f(x)| = 1\) Now we have to calculate \(f(|x|)\): Since \(0 \leq |x| \leq 2\), we'll use the second part of piecewise function \(f(x) = x - 1\). So, \(f(|x|) = |x| - 1\) Now, let's find \(g(x) = f(|x|) + |f(x)|\): \(g(x) = (|x| - 1) + 1 = |x|\) 2. When \(0 < x \leq 2\): We have \(f(x) = x - 1\), so \(|f(x)| = |x - 1|\) Now we have to calculate \(f(|x|)\): Since \(0 < |x| \leq 2\), we'll again use the second part of piecewise function \(f(x) = x - 1\). So, \(f(|x|) = |x| - 1\) Now, let's find \(g(x) = f(|x|) + |f(x)|\): \(g(x) = (|x| - 1) + |x - 1| = 2|x - 1|\) Thus, the function \(g(x)\) is given by: \(\begin{aligned} g(x) &= |x|, \quad -2 \leq x \leq 0 \\\ &= 2|x - 1|, \quad 0 < x \leq 2 .\end{aligned}\)

Step 2: Plot the graph of \(g(x)\) and determine domain and range

Plotting the graph of \(g(x)\) and analyzing it, we determine its domain and range as follows: Domain of \(g(x)\): \(-2 \leq x \leq 2\) Range of \(g(x)\): \(0 \leq g(x) \leq 2\)

Step 3: Determine the function \(\phi(x)\)

To determine the function \(\phi(x)\), we have to find \(g(|x|)\) and \(|g(x)|\), and then add them together. We'll solve this separately based on the two intervals where the piecewise function \(g(x)\) is defined. 1. When \(-2 \leq x \leq 0\): We have \(g(x) = |x|\), so \(|g(x)| = |x|\) Now we have to calculate \(g(|x|)\): Since \(0 \leq |x| \leq 2\), we'll use the second part of piecewise function \(g(x) = 2|x - 1|\). So, \(g(|x|) = 2||x| - 1|\) Now, let's find \(\phi(x) = g(|x|) + |g(x)|\): \(\phi(x) = 2||x| - 1| + |x|\) 2. When \(0 < x \leq 2\): We have \(g(x) = 2|x - 1|\), so \(|g(x)| = 2|x - 1|\) Now we have to calculate \(g(|x|)\): Since \(0 \leq |x| \leq 2\), we'll use the second part of piecewise function \(g(x) = 2|x - 1|\). So, \(g(|x|) = 2||x| - 1|\) Now, let's find \(\phi(x) = g(|x|) + |g(x)|\): \(\phi(x) = 2||x| - 1| + 2|x - 1|\) Thus, the function \(\phi(x)\) is given by: \(\begin{aligned} \phi(x) &= 2||x| - 1| + |x|, \quad -2 \leq x \leq 0 \\\ &= 2||x| - 1| + 2|x - 1|, \quad 0 < x \leq 2 .\end{aligned}\)

Step 4: Plot the graph of \(\phi(x)\) and determine domain and range

Plotting the graph of \(\phi(x)\) and analyzing it, we determine its domain and range as follows: Domain of \(\phi(x)\): \(-2 \leq x \leq 2\) Range of \(\phi(x)\): \(0 \leq \phi(x) \leq 4\)

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It measures the distance of a number from zero on the number line, ignoring any negative signs. The absolute value of a number is always non-negative. This property makes it a great tool for piecewise functions and solving real-world problems.

In this exercise, we often encounter expressions like \[ \text{|f(x)|} \text{ or } \text{|x|} \], which imply calculating the absolute value of those expressions. Here's how these calculations work:

• For any positive number, its absolute value remains the same. For instance, \( \text{|3|} = 3 \).
• For any negative number, the absolute value becomes positive. For example, \( \text{|-5|} = 5 \).
• Essentially, \( \text{|a|} = a \) if \( a \geq 0 \) and \( \text{|a|} = -a \) if \( a < 0 \).

This clean and uncomplicated nature of absolute values simplifies calculations for composite functions such as \( g(x) = f(|x|) + |f(x)| \). Understanding how and when to use these properties is crucial in function transformations and piecewise function manipulations.

Domain and Range

Understanding the domain and range of a function is crucial in analyzing and graphing functions. The domain of a function refers to all the possible input values (x-values), while the range denotes all possible output values (y-values). Knowing these two helps in sketching the graph and identifying possible limits of the function.

For the given piecewise function \( f(x) \), the domain is explicitly stated: \( -2 \leq x \leq 2 \). This means we only consider values of \( x \) within this interval. The range results from evaluating the function over its domain. In the original solution, the first derived function \( g(x)=f(|x|)+|f(x)| \) had a domain of \( -2 \leq x \leq 2 \) and a range of \( 0 \leq g(x) \leq 2 \).

These intervals help in determining what values of x lead to particular outputs. For example, because \( g(x) \) produces a range between 0 and 2, we know its output will never be negative or exceed 2 for inputs within its domain.

• The domain specifies where the function is applicable.
• The range specifies the span of outputs the function can generate.

Recognizing these aspects of functions assists in better understanding and predicting behavior on graphs.

Graphing Functions

Graphing functions allows us to visualize and analyze the behavior of mathematical expressions. It's an essential skill, especially for understanding piecewise functions and calculating solutions across different intervals. Graphs bring abstract representations of functions to life, providing clear insights into their structure and limitations.

For \( g(x) = f(|x|) + |f(x)| \), graphing involved considering how the function behaves across two specific intervals. By examining each segment separately, insights are gained into how the function transitions from one shape to another. This means sketching \( |x| \) and \( 2|x - 1| \) over their respective domains \( -2 \leq x \leq 2 \).

• Break down the function into parts if it's piecewise.
• Consider the impact of absolute values on the shape of graphs.
• Think about symmetry, especially when dealing with \( |x| \).

Graphing is accomplished by plotting points or using key function characteristics, such as intercepts and slopes, to draw each segment precisely.

This technique does not only unveil trends and patterns within the function's data but also assists in confirming the domain and range visually. Therefore, understanding the process of graphing a function from its equation and identifying key points can simplify comprehension and deepen understanding of mathematical functions.