Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function with the given properties. \(f\) has domain \([0,6]\), but is not necessarily continuous, and \(f\) attains neither a maximum nor a minimum.

Step-by-Step Solution

Answer

Sketch a function with jumps that avoids being highest or lowest anywhere on [0,6].

Step 1: Understand the Domain

We are given that the function \(f\) has a domain \([0,6]\). This means that values of \(x\) are only considered within this interval. Outside this interval, the function is not defined.

Step 2: Recognize No Continuity Requirement

The problem states that \(f\) is not necessarily continuous. This allows potential discontinuities within the interval \([0,6]\), meaning the function can have jumps or breaks.

Step 3: Avoid Attaining a Maximum or Minimum

The function \(f\) should not have a highest or lowest point within the interval, indicating it cannot have an absolute maximum or minimum at any point in its domain. This can be achieved by having discontinuous behaviors at both ends or fluctuating within the interval.

Step 4: Sketch Example Function

A way to satisfy all conditions is to design a function with one or more jumps in it. For instance, define \(f(x)\) as a step function that oscillates between two levels, say, jumping from \(1\) to \(2\) repeatedly as \(x\) moves from \(0\) to \(6\). At the endpoints, make the function incomplete, approaching but not reaching these levels, like excluding values at \(x = 0\) and \(x = 6\).

Key Concepts

Discontinuous Functions

Functions can either be continuous or discontinuous. A continuous function is smooth and unbroken over an interval. On the other hand, a discontinuous function has one or more interruptions. These interruptions are called discontinuities.
For example, a function may jump from one point to another without taking on all values in between. These jumps are a sign of discontinuity. Discontinuous points are where the limit of the function does not match the value of the function at that point.
Recognizing and plotting discontinuous functions is crucial in graph sketching. It helps to visualize how a function behaves over specific domains.

Domain of a Function

The domain of a function is the set of all possible input values (often referred to as 'x' values) for which the function is defined. When sketching graphs, understanding a function's domain is key as it tells us where to draw the graph.
If a function has a domain of \[0,6\], it means:

• The function is defined for all x-values between 0 and 6, inclusive of these points.
• We don't need to consider any x-values outside this interval, as the function does not exist there.

Knowing the domain helps in accurately sketching and interpreting the graph's behavior within the allowed range.

Maximum and Minimum Values

Maximum and minimum values of a function indicate the highest and lowest points the function reaches, respectively. These are also known as extrema. A function can have critical points where these maxima and minima occur.
However, in certain cases, a function may not attain a true maximum or minimum, especially if it is discontinuous.
In the current context, it is important to note that we want our function not to attain a maximum or minimum value over its defined domain. This means our function should oscillate or have jumps without settling at a highest or lowest point.

Step Function

A step function is a special type of piecewise function that stays constant over specific intervals and then jumps to a different value. This creates a graph that looks like a series of steps.
Step functions are an excellent way to represent discontinuous functions that do not have a maximum or minimum value at a specific point.
In graph sketching exercises, functions may jump abruptly between values. This can help us model a function that doesn't possess absolute extrema. In this scenario, a simple example is when a function jumps between values 1 and 2 continuously over a defined domain, ensuring it does not settle at a particular maximum or minimum.