Problem 6 - Calculus for Biology and Medicine

Question

(a) Show that $$ 2|x-2|=\left\\{\begin{array}{ll} 2(x-2) & \text { for } x \geq 2 \\ 2(2-x) & \text { for } x \leq 2 \end{array}\right. $$ (b) Are the functions \(f(x)\) and \(g(x)\) equal? $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} 4-2 x & \text { for } 0 \leq x \leq 2 \\ 2 x-4 & \text { for } 2 \leq x \leq 3 \end{array}\right. \\ g(x)=2|x-2|, x \in[0,3] \end{array} $$

Step-by-Step Solution

Answer

(a) Verified using absolute value properties. (b) Yes, functions are equal.

Step 1: Understand the Given Function

The exercise provides the function \(2|x-2|\) and its piecewise expression. We need to demonstrate the provided result by considering the definition of absolute value.

Step 2: Express Absolute Value for Different Intervals

The absolute value function \(|x-2|\) can be broken down into two cases: 1. \(x-2\) when \(x \geq 2\) 2. \(-(x-2) = 2-x\) when \(x < 2\)

Step 3: Substitute into the Original Equation

Substitute each case into \(2|x-2|\):- For \(x \geq 2\): \(2|x-2| = 2(x-2)\)- For \(x < 2\): \(2|x-2| = 2(2-x)\)

Step 4: Verify the Given Expression

Match the substitutions with the given function:- \(2(x-2)\) corresponds to \(x \geq 2\)- \(2(2-x)\) corresponds to \(x \leq 2\), which covers \(x < 2\) as well, since it is inclusive of the boundary.

Step 5: Evaluate if \(f(x)\) and \(g(x)\) are Equal

Check intervals and expressions for \(f(x)\) and \(g(x)\):- For \(0 \leq x \leq 2\): - \(g(x) = 2(2-x) = 4 - 2x\), potentially equals \(f(x) = 4 - 2x\)- For \(2 \leq x \leq 3\): - \(g(x) = 2(x-2) = 2x - 4\), matches \(f(x) = 2x - 4\)- Both functions are defined over \([0, 3]\) and have matching expressions in their respective intervals.

Key Concepts

Absolute Value Function

The absolute value function is a special kind of function that measures distance. In mathematics, it tells us how far a number is from zero, regardless of its direction. This is particularly useful when dealing with piecewise functions like in our given problem where we analyze expressions based on variable conditions.

For example, the absolute value of \[|x-2|\] is really just saying "How far away is \(x\) from 2?". Depending on whether \(x\) is greater than, less than, or equal to 2, this distance will change slightly in representation.

• If \(x \geq 2\), then the distance, or \(|x-2|\), simply becomes \(x-2\).
• If \(x < 2\), then \(|x-2|\) transforms into \(-(x-2)\), which is the same as \(2-x\).

Understanding this, we can see that the absolute value function lets us break down formulas considering different conditions, like using piecewise notation.

Intervals

Intervals are a way of expressing a range of numbers. They give us a clear picture of where a particular rule or formula applies when dealing with functions. In math, particularly in this problem, we use intervals to manage piecewise functions which behave differently in different sections of their domain.

• The interval \([0, 3]\) tells us that we're working with numbers starting from 0 all the way up to 3, inclusive.
• For the function \(2|x-2|\), this function is further broken down into two conditions based on the intervals: \(x \geq 2\) and \(x < 2\).

This use of intervals helps set boundaries for our calculations and clarifies conditions for piecewise functions.

Function Equality

Function equality revolves around the idea that two functions can be considered the same if they yield identical results for every possible input within their defined domains. In the context of the given problem, we need to verify if functions \(f(x)\) and \(g(x)\) are equal across their given interval.

Analyzing each function:

• For the interval \([0, 2]\), both functions \(f(x) = 4 - 2x\) and \(g(x) = 2(2-x)\) result in the same expression, thus equating them.
• From \([2, 3]\), both \(f(x) = 2x - 4\) and \(g(x) = 2(x-2)\) match as well.

When both functions yield the same outcome for every \(x\) in the interval \([0, 3]\), they are considered equal. Function equality is essential in confirming the relationship between two functions based on their specific expressions and intervals.