Problem 6 - Contemporary Precalculus

Question

Sketch the graph of the function, being sure to indicate which endpoints are included and which ones are excluded. $$f(x)=-[x]$$

Step-by-Step Solution

Answer

Question: Given the function $$f(x) = -[x]$$, where $$[x]$$ represents the greatest integer less than or equal to $$x$$, sketch the graph of the function, focusing on the integer intervals and the breaks between them, and indicating the included/excluded endpoints and discontinuities. Answer: Study the function $$f(x) = -[x]$$ within integer intervals and observe its behavior and properties, such as inclusions/exclusions and discontinuities. Sketch the graph considering the function behavior in integer intervals, including left endpoints and excluding right endpoints, and showing the discontinuities at integer values.

Step 1: Understand the function and its properties

The function in question is $$f(x) = -[x]$$, where $$[x]$$ represents the greatest integer less than or equal to $$x$$. This means that we should focus on finding the values of $$f(x)$$ in the integer intervals and the breaks between them. Step 2: Analyze the function behavior within integer intervals

Step 2: Evaluate the function for integer intervals

Since $$[x]$$ is constant within integer intervals, we'll find the values of the function in integer intervals. For example, let $$x$$ be in the interval $$(0, 1)$$, then, $$[x] = 0$$ and $$f(x) = -[x] = -0 = 0$$. Similar evaluations can be done for other integer intervals. Step 3: Determine the endpoints of the intervals

Step 3: Find endpoints and their inclusion/exclusion

To identify if the endpoints are included or excluded, we need to evaluate the function at integer values. Let's evaluate $$f(x)$$ at an integer point $$n$$, then, $$f(n) = -[n] = -n$$. Since this is a valid value for the function, we can conclude that the left endpoints of each interval are included. Since right endpoints merge with the left endpoints of the next interval, they are not included. Step 4: Identify possible discontinuities

Step 4: Check for function discontinuity

Given how the function is defined, we know that $$f(x)$$ tends to have discontinuities at integer values. Therefore, in our graph, we will show breaks at each integer value. Step 5: Sketch the graph

Step 5: Draw the graph of the function

Using the information obtained in the previous steps, sketch the graph of $$f(x) = -[x]$$ ensuring that you accurately portray the behavior of the function in integer intervals, the inclusion of left endpoints and exclusion of right endpoints, and the discontinuity of the function at integer values.

Key Concepts

Greatest Integer Function

The greatest integer function, sometimes called the floor function, is a piecewise function that is fundamental in understanding the mechanics behind step functions. It is denoted by \( [x] \), where for each real number \( x \), the greatest integer function outputs the largest integer less than or equal to \( x \). Imagine sliding a number down a number line until it hits the nearest integer below or directly on it; this is the principle of the greatest integer function.

For example, for \( x = 2.7 \), the greatest integer function will be \( [2.7] = 2 \), because 2 is the greatest integer less than or equal to 2.7. Similarly, \( [3] = 3 \), as 3 is an integer itself. The greatest integer function creates 'steps' in the graph, hence its categorization as a step function.

An important aspect of graphing these functions is to illustrate this stepping behavior accurately. When graphing \( f(x) = -[x] \), the negative sign indicates that the 'stairs' of our graph will be descending, not ascending as they do for the pure \( [x] \) function.

Function Discontinuity

A function discontinuity occurs when there is an abrupt change in the value of the function. In the context of the greatest integer function, and consequently for \( f(x) = -[x] \), discontinuities arise at every integer point. The function does not proceed smoothly from one value to the next; instead, it 'jumps'.

The point where the function jumps is called a discontinuity, and these are crucial when plotting the graph of a step function. If \( x \) is approaching an integer value from the left, the function value will suddenly jump to the value of the greatest integer at that point. This is because the greatest integer function steps up to the next integer as soon as \( x \) crosses an integer value.

To capture this in the graph, you would draw open circles at the points where \( x \) is an integer to indicate the value of the function there does not include that point. Remember, each jump in value creates a separate interval on the graph, showing clear separation between different function values.

Endpoints Inclusion and Exclusion

Dealing with endpoints inclusion and exclusion is critical when graphing functions such as \( f(x) = -[x] \). Here's what you need to remember: the left endpoint of each interval, the point where the interval begins, is included in the function. This is depicted graphically by a filled-in dot. On the other hand, the right endpoint is excluded since it is the starting point of a new interval. Graphically, an open circle denotes this exclusion.

In the case of our function \( f(x) = -[x] \), each interval's left endpoint, which is an integer, reflects a filled-in dot at that integer. This inclusion signifies that the function indeed takes the value of \( -[x] \) at that point. In contrast, each interval's right endpoint, at the next integer, would be open to signify the step up to a new 'stair'. This captures the precise nature of the function's behavior at every integer value, ensuring each step is distinctly represented on the graph.