Problem 6 - Calculus for Biology and Medicine

Question

(a) Show that $$ 2|x-1|=\left\\{\begin{array}{ll} 2(x-1) & \text { for } x \geq 1 \\ 2(1-x) & \text { for } x \leq 1 \end{array}\right. $$ (b) Are the functions $$ f(x)=\left\\{\begin{array}{ll} 2-2 x & \text { for } 0 \leq x \leq 1 \\ 2 x-2 & \text { for } 1 \leq x \leq 2 \end{array}\right. $$ and $$ g(x)=2|x-1|, x \in[0,2] $$ equal?

Step-by-Step Solution

Answer

Yes, the function \(f(x)\) is equal to \(g(x)\) for \(x \in [0, 2]\).

Step 1: Understanding Absolute Value

Recall that the definition of absolute value states: \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). We can therefore express the absolute value \(|x-1|\) using this definition for different values of \(x\).

Step 2: Expressing Absolute Value as a Piecewise Function

Let's express \(|x-1|\) given x relative to 1:- When \(x \geq 1\), \(x-1 \geq 0\), so \(|x-1| = x - 1\).- When \(x < 1\), \(x-1 < 0\), so \(|x-1| = -(x-1) = 1 - x\).Now substituting back into the original expression, \(2|x-1| = 2(x-1)\) for \(x \geq 1\) and \(2|x-1|=2(1-x)\) for \(x < 1\).

Step 3: Verify Equality for Functions f(x) and g(x)

We are given \(f(x)\) and \(g(x) = 2|x-1|\) for \(x \in [0,2]\). Comparing pieces:1. For \(0 \leq x \leq 1\), \(f(x) = 2 - 2x\) and \(g(x) = 2(1-x) = 2 - 2x\), both are equal.2. For \(1 \leq x \leq 2\), \(f(x) = 2x - 2\) and \(g(x) = 2(x-1) = 2x - 2\), both are equal.Since both functions are equal for each piece in the respective intervals, \(f(x) = g(x)\) for \(x \in [0, 2]\).

Key Concepts

Absolute Value

The concept of absolute value is centered around the idea of measuring distance from zero on a number line, regardless of direction. This means that the absolute value of a number is always non-negative. Mathematically, the absolute value \[|x|\]is defined as:

• \(x\) if \(x \geq 0\)
• -\(x\) if \(x < 0\)

Applying this definition to \(|x-1|\),we must consider two cases based on the value of \(x\).
If \(x\) is greater than or equal to 1, then \(x - 1\) is non-negative, so \(|x - 1| = x - 1\).
If \(x\) is less than 1, then \(x - 1\) is negative, making \(|x - 1| = -(x - 1) = 1 - x\).This demonstrates how the sign of the expression changes the result of the absolute value, providing insights that are crucial when writing piecewise definitions of functions involving absolute values.

Function Equality

Function equality occurs when two functions yield the same output for every possible input in their common domain. When comparing functions, \(f(x)\)and \(g(x)\),it is essential to ensure they provide identical results across all values they are defined. For example, comparing the piecewise function\[f(x) = \left\{\begin{array}{ll}2-2x & \text{for } 0 \leq x \leq 1 \2x-2 & \text{for } 1 \leq x \leq 2\end{array}\right.\]with \(g(x) = 2|x-1|\) over the interval \([0,2]\),we verify equality by checking two key intervals:

• For \(0 \leq x \leq 1\), \(f(x)\) yields \(2 - 2x\),and \(g(x)\) reforms to \(2(1-x) = 2 - 2x\),proving equality.
• For \(1 \leq x \leq 2\), \(f(x)\) becomes \(2x - 2\),and \(g(x)\) also results in \(2(x-1) = 2x - 2\),confirming equal outputs.

Thus, these two functions are equal since they are identical for each part of the respective intervals. Understanding function equality can help simplify problem-solving and ensure accuracy across mathematical proofs.

Piecewise Function Definition

Piecewise functions are defined by different expressions depending on the interval of the input. Often used in mathematics to represent functions with varying behavior in different regions, a piecewise function partitions its domain into intervals and assigns an expression to each. For instance, consider the function\[2|x-1| = \left\{\begin{array}{ll}2(x-1) & \text{for } x \geq 1 \2(1-x) & \text{for } x < 1\end{array}\right.\]
This function changes its formula based on whether \(x\)is below or above 1, each part derived from how the absolute value \(|x-1|\)behaves.

• If \(x \geq 1\),use \(2(x-1)\).
• If \(x < 1\),use \(2(1-x)\).

Piecewise functions provide a powerful tool for modeling real-world situations where a single rule for a function isn’t sufficient. Writing piecewise definitions involves understanding the intervals where each expression is valid, ensuring there are no gaps or overlaps between the cases.