Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function that has domain \([0,2]\) and is continuous on \([0,2)\) but not on \([0,2]\).

Step-by-Step Solution

Answer

Sketch \(f(x) = x\) for \(x \in [0, 2)\) with \(f(2) = 1\).

Step 1: Understand the Domain and Continuity Requirements

The function must be defined for all values in the interval \([0, 2]\) and remain continuous up to but not including the endpoint \(x = 2\). At \(x = 2\), the function should have a point of discontinuity.

Step 2: Sketch a Continuous Segment on [0, 2)

Begin by sketching a graph on the interval \([0, 2)\) that is continuous. A simple example could be the function \(f(x) = x\), which is a linear function. Draw a line from \(x = 0\) to just before \(x = 2\) without including \(x = 2\).

Step 3: Introduce Discontinuity at x = 2

To introduce the discontinuity at \(x = 2\), you could define the function just outside the continuous behavior. For instance, let \(f(2) = 1\), a different value than would be expected from the continuity in \([0, 2)\). Represent this with a point on the graph at the coordinate \((2, 1)\) that does not connect with the line from \([0, 2)\).

Step 4: Verify Continuity and Discontinuity

Check that as \(x\) approaches \(2\) from the left, the function appears continuous (e.g., it approaches the expected value if \(f(x) = x\), approaching \(f(2) = 2\)), but at \(x = 2\) the function value changes to \(f(2) = 1\). This confirms discontinuity at \(x = 2\).

Key Concepts

Graphing

Graphing helps us visually interpret and analyze the behavior of functions. When tackling an exercise like this, sketching provides a clear understanding of how the function behaves across its domain. For functions such as this one, with a specific continuity and discontinuity characteristic, start by drafting the section that is supposed to be continuous first.

• Identify what portions of the domain the function should remain continuous. Our given exercise specifies this for the interval \([0, 2)\).
• Draw the appropriate segment for the function, which could be as simple as a straight line, or a curve, depending on the function's nature.
• In our case, a line representing\(f(x) = x\) from \(x = 0\) to just before \(x = 2\) effectively captures the continuous part.

A graph not only shows where the function is continuous or discontinuous but also offers an immediate visual clue to any unexpected changes in behavior, making it an indispensable tool in any mathematical analysis.

Domain

The domain is fundamental in defining the "universe of discourse" of a function, meaning it describes all possible values of \(x\) for which the function is defined. In our exercise, the function has a well-defined domain of \([0, 2]\). This includes every real number from 0 up to and including 2.

• Consider \([0, 2)\) as the portion where the function is continuous, meaning that every value in this interval naturally leads to its respective function output.
• The domain also includes the endpoint \(x = 2\), but as specified, there's a discontinuity right at this point.

Being mindful of the domain helps in understanding the possible inputs for the function and relates directly to its continuity and discontinuity. With these details established, the function is entirely described and its behavior over the defined interval becomes predictable.

Continuity

Continuity in functions describes how seamlessly one can move from one point to another within a function's domain without interruption. This is crucial in many mathematical applications, as continuous functions ensure predictability and smooth transitions.

• For a function to be continuous on an interval like \([0, 2)\), it must smoothly connect every value in that interval without any sudden jumps or breaks.
• In our example, using \(f(x) = x\) illustrates this as we start with \(x = 0\) and progresses linearly to just below \(x = 2\), without deviation from the expected path.

Whereas continuity is natural in many real-world situations, in some cases, like our exercise, it intentionally leading up to a disruption near the domain's endpoint. Understanding and identifying these transitions helps in predicting how a function may behave just before it hits a discontinuity.

Discontinuity

Discontinuity represents a break, gap, or jump in the graph of a function, making it non-connected at a point or over an interval. Identifying discontinuities is crucial to understanding exceptions where the function's behavior differs from its typical pattern.

• In this exercise, the discontinuity occurs exactly at \(x = 2\), where the function \(f(x)\) should not "follow through" smoothly from the rest of the interval \([0, 2)\).
• Specifically, we assign \(f(2) = 1\), creating a deliberate divergence from the expected value (which would be \(2\) if it were continuous).

This apparent disconnect signifies a step away from continuity and serves as a clear example of how a single point can alter an entire interval's behavior. Recognizing such discontinuities is central to comprehensively grasping a function's complete nature.