Problem 23 - Single Variable Calculus

Question

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ f(x)=\left\\{\begin{array}{ll} x & \text { if }-1 \leq x \leq 0 \\ 2-x & \text { if } 0

Step-by-Step Solution

Answer

The graph of the function consists of two linear segments, with \(f(x) = x\) for \(-1 \leq x \leq 0\) and \(f(x) = 2-x\) for \(0 < x \leq 2\). The absolute minimum value is \(f(x) = -1\) at \(x = -1\) and the absolute maximum value is \(f(x) = 2\) at \(x = 0\).

Step 1: Sketch the graph of the function

First, we need to plot both parts of the function and observe their behavior. For the first part, \(f(x) = x\) for \(-1 \leq x \leq 0\): This is a linear function with a slope of \(1\) and y-intercept at \(0\). Plot the graph for the given range. For the second part, \(f(x) = 2-x\) for \(0 < x \leq 2\): This is also a linear function, but with a slope of \(-1\) and y-intercept at \(2\). Plot this graph for the given range. When sketching the graph, keep in mind that for \(x=0\), the first equation is valid, and then the second equation takes over. In the end, we will have connected graphs for both parts of the function.

Step 2: Find the absolute maximum and minimum

To find the absolute maximum and minimum values, we need to check the endpoints of each part of the function and any critical points within the given range. For \(f(x) = x\) in the range \(-1 \leq x \leq 0\), the function is strictly increasing with no critical points. Therefore, we only need to check the endpoints: - At \(x = -1\), \(f(x) = -1\). - At \(x = 0\), \(f(x) = 0\). For \(f(x) = 2-x\) in the range \(0 < x \leq 2\), the function is strictly decreasing with no critical points. Therefore, we only need to check the endpoint: - At \(x = 2\), \(f(x) = 0\). Now, compare these values to find the absolute maximum and minimum: - Absolute minimum value: \(f(x) = -1\), at \(x = -1\) - Absolute maximum value: \(f(x) = 2\), at \(x = 0\).

Key Concepts

Absolute Maximum

In mathematical terms, an absolute maximum is the highest point, or the greatest value, a function reaches within a given domain. When dealing with piecewise functions, like in this case, finding the absolute maximum requires analyzing individual segments and endpoint values.

For the provided function, we observe that the function is divided into two linear segments. The first segment \( f(x) = x \) ranges between \( -1 \) and \( 0 \), and naturally, its highest value is at one of its endpoints. Similarly, for the second segment \( f(x) = 2-x \) ranging from \( 0 \) to \( 2 \), we evaluate its endpoints as well.

Looking at each of these segments:

• For the part \( x \) in the interval \(-1 \leq x \leq 0\), \( f(x) = 0 \) at \( x = 0 \), which is the highest point for this segment.
• For the part \( 2-x \) in the interval \( 0 < x \leq 2\), the endpoint at \( x = 0 \) also yields the value \( f(x) = 2 \), which becomes the absolute maximum of the whole function.

Overall, evaluating each diagram point ensures accurate determination of the absolute maximum, which is \( 2 \) when \( x = 0 \). This process is crucial in graph sketching of piecewise functions.

Absolute Minimum

The absolute minimum of a function refers to the smallest value that it takes within a specific domain. Understanding the absolute minimum is essential when dealing with different segments of piecewise functions.

In our example, the piecewise function is composed of linear sections with their respective domains. To find the absolute minimum:

• We look at the segment \( f(x) = x \), where for \( x = -1 \), the function reaches \( f(x) = -1 \), which is the lowest value in this domain.
• The other segment \( f(x) = 2-x \), doesn't provide a smaller value than \( -1 \) within its range, since even at its endpoints the outcomes remain non-negative.

By comparing all endpoint values across the disjoint segments, it's clear that the absolute minimum for the entire function is \( -1 \). Identifying such a minimum efficiently aids in visualizing the function's behavior and representing it correctly on a graph.

Graph Sketching

Sketching graphs of piecewise functions involves understanding each segment's behavior and how they connect at the domain boundaries. The aim is to create a clear depiction of each part of the function using the points and line slopes given.

Let's break down the steps:

• First, analyze the part where \( f(x) = x \) for \( -1 \leq x \leq 0 \). This is a line with a slope of \( +1 \), starting from \( (-1, -1) \) and going up to \( (0, 0) \).
• Next, consider the segment \( f(x) = 2-x \) where \( 0 < x \leq 2 \). This has a slope of \( -1 \) and it goes from the point just after zero up to \( (2, 0) \).

Remember that at \( x=0 \), the graph will join neatly because the first function terminates exactly at \( (0, 0) \), where the second function begins from \( (0, 2) \). Ensure to mark any open circles at endpoints when segments don't include the boundary point. Following these steps, drawing a proportionate sketch of a piecewise function becomes a structured process.

Piecewise Linear Function

Piecewise linear functions are functions defined by multiple linear expressions, each valid over a specific interval. This structure allows for a precise representation of segments on a graph where straightforward relationships are inadequate.

In the context of our function:

• The first linear expression \( f(x) = x \) describes a line from \( -1 \) to \( 0 \) with a positive slope, indicating its upward nature within that interval.
• The second, \( f(x) = 2-x \), defines a downward-sloping line taking over exactly after \( 0 \) and ending at \( x = 2 \).

Handling piecewise linear functions involves combining these line segments accordingly, aligning them at their borders within respective boundaries. This is particularly useful in scenarios such as scheduling, economics, and other fields where changes occur in a stepwise manner. Understanding this concept is critical for accurately analyzing and presenting these functions in a graphical form.