Problem 81 - College Algebra with Modeling and Visualization

Question

Exercises \(79-82:\) Complete the following. (a) Use dot mode to graph the function \(f\) in the standard viewing rectangle (b) Evaluate \(f(-3.1)\) and \(f(1.7)\) $$ f(x)=2[x]+1 $$

Step-by-Step Solution

Answer

Graph consists of horizontal segments. \( f(-3.1) = -7 \); \( f(1.7) = 3 \).

Step 1: Understanding the Function

The function given is \( f(x) = 2[x] + 1 \), where \([x]\) is the greatest integer less than or equal to \( x \). This is also known as the floor function.

Step 2: Sketching the Graph

In the "dot mode" you would plot individual points that form the graph. For the function \( f(x) = 2[x] + 1 \), the plot will consist of horizontal line segments. The function changes value at every integer point. For instance, points may include: \((0, 1)\), \((0.9, 1)\), \((1, 3)\), \((1.9, 3)\), \((2, 5)\), and so on. Each segment from \( x = n \) to \( x = n+1 \) is horizontal, and at the "jump," the function steps to a higher level.

Step 3: Evaluate f(-3.1)

To evaluate \( f(-3.1) \), identify the greatest integer less than or equal to \(-3.1\), which is \(-4\). Substitute this value into the function: \( f(-3.1) = 2(-4) + 1 = -7\).

Step 4: Evaluate f(1.7)

Similarly, for \( f(1.7) \), find the greatest integer less than or equal to \(1.7\), which is \(1\). Substitute this value into the function: \( f(1.7) = 2(1) + 1 = 3\).

Key Concepts

Greatest Integer Function

The greatest integer function is a specific type of mathematical function that assigns to each real number the largest integer less than or equal to it. This function is also known as the floor function and is represented as \([x]\). For example, the greatest integer value of 3.7 is 3, and the greatest integer for -2.6 is -3. This is because 3 is the largest whole number less than or equal to 3.7, and -3 is the largest whole number less than or equal to -2.6.

Understanding this function is crucial because it forms the basis for many real-world applications where rounding down is necessary. It's important to remember that the function \([x]\) is a step function, meaning its value only changes at integer points. With this in mind, it’s easier to understand how functions such as \(f(x) = 2[x] + 1\) operate, where each step is defined by the integer boundary acting as the transition point for the value of \(f(x)\).

Applying it in evaluations involves identifying the greatest integer less than or equal to a given number, then using it in subsequent mathematical operations effectively, as demonstrated in the exercise.

Graphing Piecewise Functions

Graphing piecewise functions can often seem challenging, but once broken down, it's quite manageable.

A piecewise function is defined by different expressions in different parts of its domain. The greatest integer function is inherently piecewise, as it features segments of constant value separated by jumps at integer points. When graphing a function like \(f(x) = 2[x] + 1\), the graph consists of horizontal line segments that represent the function’s value between integer inputs.

• Each segment covers the interval between consecutive integers, such as from \[n\] to \[n+1\].
• There’s a jump at each integer value, where the function steps up.

When plotting in dot mode, you’ll see discrete points marking these steps. For engineering the graph, it helps to plot key points like \(0,1\), \(1,3\), and then connect them with horizontal lines, changing levels at each integer. This approach helps visually convey the nature and behavior of the function at these crucial points.

Evaluating Functions

Evaluating functions is a fundamental skill in mathematics. It involves finding the value of a function for specific inputs. For the exercise at hand, evaluating \(f(-3.1)\) and \(f(1.7)\) requires knowledge of how the greatest integer function operates.

First, determine the greatest integer less than or equal to the given input:

• For \(f(-3.1)\), the greatest integer less than \(-3.1\) is \(-4\).
• For \(f(1.7)\), the greatest integer less than \(1.7\) is \(1\).

Next, substitute these integers into the function expression \(f(x) = 2[x] + 1\) and perform the arithmetic:

• For \(f(-3.1) = 2(-4) + 1 = -7\),
• For \(f(1.7) = 2(1) + 1 = 3\).

These calculations demonstrate the systematic method of evaluating step functions, particularly those using the greatest integer concept, and highlight the importance of accurate integer identification in mathematical expressions.