Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=2[x \rrbracket\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 2 \lfloor x \rfloor\) will resemble a staircase, with steps at each integer value of x. This is due to the floor function rounding down x to the nearest integer and subsequently multiplying it by 2.

Step 1: Understand the function

The given function is \(f(x) = 2 \lfloor x \rfloor\). The floor function will round x down to the nearest integer, and this result is then multiplied by 2.

Step 2: Calculate sample points

Choose a variety of x-values, both positive and negative, as well as some decimal values, to showcase how the floor function behaves. For the purpose of this problem, let's choose x-values of -2.5, -1, 0, 1.5, and 3. Their corresponding y-values will be \( f(-2.5) = 2 \lfloor -2.5 \rfloor = -6 \), \( f(-1) = 2 \lfloor -1 \rfloor = -2 \), \( f(0) = 2 \lfloor 0 \rfloor = 0 \), \( f(1.5) = 2 \lfloor 1.5 \rfloor = 2 \), and \( f(3) = 2 \lfloor 3 \rfloor = 6 \).

Step 3: Sketch the graph

Now that certain points have been determined ([-2.5, -6], [-1, -2], [0, 0], [1.5, 2], [3, 6]), plot them on an x-y grid. Connect these points with lines to indicate the change between each integer point. This will result in a step-like graph. Each 'step' corresponds to an integer value of x, repeating until the next integer is reached.

Key Concepts

Step Function

A step function is a special kind of function heavily used in mathematics and applied sciences. Its graph looks like a series of flat, horizontal lines or steps, which gives it its famous name.
In mathematical terms, a step function is usually defined with the help of the floor function, like in our example with the function \(f(x)=2\lfloor x\rfloor\). This function takes any real number input and rounds it down to the nearest integer. The result is then multiplied by two.
Step functions are quite fascinating because they do not increase or decrease linearly. Instead, they stay constant over an interval and then "jump" to a new value at certain points. This type of function is useful for modeling situations where quantities change abruptly rather than gradually. Because of this property, many students find it helpful to visualize step functions on a graph to better understand how they work.

Plotting Points

Plotting points on a graph is a fundamental skill in mathematics. When dealing with functions like step functions, it becomes essential to choose appropriate x-values to see how the function behaves.
For the function \(f(x) = 2\lfloor x\rfloor\), it helps to choose a mix of whole numbers and decimals to illustrate the rounding effect of the floor function. Start by selecting different kinds of numbers such as:

• Negative numbers, like -2.5
• Zero, as a neutral benchmark
• Positive integers, like 1 or 3
• Decimal points, such as 1.5

This selection provides a full range of the floor function's behavior. After calculating the corresponding y-values, you can accurately plot these points on the graph, helping to showcase the graph's stair-like nature.

Integer Rounding

Integer rounding is the process of rounding a real number down to the nearest whole number. In our function \(f(x) = 2\lfloor x\rfloor\), we use integer rounding through the floor function, denoted by \(\lfloor x\rfloor\). The floor function takes a real number and rounds it down to the closest integer that is not greater than the number itself.
For example:- \(\lfloor -2.5 \rfloor\) gets rounded down to -3- \(\lfloor 1.5 \rfloor\) becomes 1
Understanding this concept is vital when dealing with functions like step functions because it dictates how the graph "steps" from one level to the next. This principle demonstrates how non-linear changes in data can be modeled efficiently using simple mathematical techniques.

Graph Interpretation

Interpreting a graph is about understanding its visual representation to gain insights into the function's behavior. With a step function, the graph tells you a lot about how the function works, even at a glance.
When you look at the step-like structure of \(f(x) = 2\lfloor x\rfloor\), you can see how the function jumps between constant values at specified intervals. Each horizontal segment represents the function's output remaining constant over an interval until it encounters an integer.
This type of graph is particularly useful for analyzing real-world situations where values stay constant over a period, like utility rates or tax brackets. Being able to draw conclusions from the shape and structure of a step function graph is essential for applying these concepts in both academic and practical scenarios. Read the graph carefully, paying attention to the "steps," to fully grasp the discrete nature of the function's output.