Problem 122 - Calculus for JEE ADVANCED

Question

Let \(\begin{aligned} f(x)=& x+2|x+1|+2|x-1| \\ &= \begin{cases}x-2(x+1)-2(x-1), & x<-1 \\ x+2(x+1)-2(x-1), & -1 \leq x \leq 1 \\\ x+2(x+1)+2(x-1), & x>1\end{cases} \\ &= \begin{cases}-3 x, & x<-1 \\\ x+4, & -1 \leq x \leq 1 \\ 5 x, & x>1\end{cases} \end{aligned}\)

Step-by-Step Solution

Answer

The piecewise function is \( f(x) = \begin{cases} -3x & x < -1 \\ x+4 & -1 \leq x \leq 1 \\ 5x & x > 1 \end{cases} \).

Step 1: Understand Piecewise Definition

The function \( f(x) \) is expressed in a piecewise definition based on conditions related to \( x \). This means the form of the function changes based on the interval in which \( x \) lies. We need to determine which case applies by evaluating the condition on \( x \).

Step 2: Analyze Case 1: \( x < -1 \)

For \( x < -1 \), we substitute \( x \) into the function expression: \( f(x) = x - 2(x+1) - 2(x-1) \). Simplifying this, we calculate:\( = x - 2x - 2 - 2x + 2 \)\( = -3x \).So, for \( x < -1 \), \( f(x) = -3x \).

Step 3: Analyze Case 2: \( -1 \leq x \leq 1 \)

For \( -1 \leq x \leq 1 \), substitute \( x \) into the expression:\( f(x) = x + 2(x+1) - 2(x-1) \). Simplifying this:\( = x + 2x + 2 - 2x + 2 \)\( = x + 4 \).In this interval, \( f(x) = x + 4 \).

Step 4: Analyze Case 3: \( x > 1 \)

For \( x > 1 \), substitute \( x \) into the expression:\( f(x) = x + 2(x+1) + 2(x-1) \). Simplifying:\( = x + 2x + 2 + 2x - 2 \)\( = 5x \).Thus, for \( x > 1 \), \( f(x) = 5x \).

Step 5: Formulate the Complete Piecewise Function

Having analyzed each interval, the entire piecewise function can be consolidated as:the piecewise function:\[ f(x) = \begin{cases} -3x & \text{if } x < -1 \ x+4 & \text{if } -1 \leq x \leq 1 \ 5x & \text{if } x > 1 \end{cases} \]

Key Concepts

Understanding Absolute Value

The concept of absolute value is fundamental to mathematics. It represents the distance of a number from zero on the number line, without considering direction. This means that absolute value always results in a non-negative number. The absolute value function is denoted by two vertical bars: \( |x| \). For example:

• \( |3| = 3 \)
• \( |-3| = 3 \)

This property makes absolute value useful for dealing with quantities that only relate to size and not direction, such as distance. When dealing with expressions like \( |x+1| \) and \( |x-1| \), you assess whether the values inside the absolute value symbols are positive or negative, based on your chosen interval for \( x \). The formula \( |a| = a \) if \( a \geq 0 \) and \( |a| = -a \) if \( a < 0 \) assists in unfolding expressions involving absolute values.

Exploring Function Intervals

The concept of dividing a function into intervals helps evaluate how the function behaves under different conditions. For a piecewise function like \( f(x) \), different mathematical expressions define the function in each interval. Determine which interval an \( x \) value falls into, as this changes the formula for \( f(x) \). For example, given our piecewise function:

• For \( x < -1 \), we use \( -3x \)
• For \( -1 \leq x \leq 1 \), we use \( x+4 \)
• For \( x > 1 \), we use \( 5x \)

Intervals indicate distinct parts or segments where the function's behavior changes. Understanding these ranges and implementing the correct expression is crucial for accurately graphing or analyzing the function's properties.

Simplifying Expressions

Simplifying expressions is a vital skill in mathematics. It involves reducing expressions to their simplest form. This process makes them easier to work with and understand.In our piecewise function exercise, you start by substituting specific \( x \) values into the function and performing operations within each interval.Consider the expression \( x - 2(x+1) - 2(x-1) \):

• First, distribute the \( -2 \) across \( (x+1) \) and \( (x-1) \)
• Combine like terms, leading to \( -3x \)

This results in a simpler expression that is much easier to analyze and use. Remember, when simplifying expressions containing absolute values and multiple terms, each step should clearly adhere to mathematical rules to maintain accuracy and meaning.