Problem 21 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=3[x]\)

Step-by-Step Solution

Answer

The function is a greatest integer (step) function. Its graph includes horizontal segments that step up every integer.

Step 1: Identify the function type

The given equation is \(y = 3[x]\). The notation \([x]\) indicates the greatest integer function, also known as the floor function. This function outputs the greatest integer less than or equal to \(x\). Therefore, our equation represents a piecewise function modified by a constant multiplier.

Step 2: Rewrite as a piecewise function

Since \([x]\) stands for the greatest integer less than or equal to \(x\), we can express the function as \(y = 3n\) where \(n\) is the greatest integer before \(x\). For example, if \(2.3 \leq x < 3\), then \([x] = 2\) and thus \(y = 3 \times 2 = 6\).

Step 3: Determine key points

Let's find key points values to help in graphing. Use various \(x\) to calculate \(y\): - For \(x = 0\), \([x] = 0\), \(y = 0\).- For \(x = 1.5\), \([x] = 1\), \(y = 3\).- For \(x = 3\), \([x] = 3\), \(y = 9\). These show how the function steps at integers.

Step 4: Graph the function

Plot the points derived from step 3. The graph will appear as horizontal line segments over each integer range: the line segments will be horizontal until the next integer where they will jump in value, illustrating a step function. Each segment is 3 units higher than the previous.

Key Concepts

Understanding Step Functions

A step function is a function that remains constant within certain intervals and then makes a sudden jump to a different value at specific points, akin to steps. This gives rise to its name, as it appears like a series of steps when graphed. One key feature of step functions is that they do not connect continuously between these jumps.

Typically, in a step function, each piece or section resembles a flat line segment, until it reaches a point where it abruptly shifts vertically to another level. Here are some salient points about step functions:

• They are often defined using other mathematical expressions, such as the greatest integer function.
• Each "step" occurs at integer boundaries in many typical examples.
• The graph is a series of horizontal segments indicating constant intervals.

Step functions are essential in mathematical modeling when sudden changes in conditions are represented.

Exploring Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each with their own domain. In essence, the graph of a piecewise function combines various segments, each representing a separate segment of the calculation over different intervals. This allows for versatile mathematical models in a wide range of scenarios.

Key characteristics of piecewise functions include:

• Different rules or formulas are applied to different intervals of the domain.
• The graph can be a combination of different types of segments, including straight lines, curves, or steps.
• Each segment is disconnected from the others at specific cutoff points.

This flexibility is valuable for functions like our given equation, where multiplying the step function by a constant creates different levels over the domain.

Demystifying the Floor Function

The floor function, symbolized as \([x]\), captures the essence of finding the greatest integer less than or equal to a given number \(x\). It's a crucial mathematical construct, especially useful in rounding down operations. The floor function is often confused with simply rounding down, but its distinctive property of only moving to the previous integer level makes it unique and precise in its operation.

Here’s why the floor function is special:

• For any real number \(x\), \([x]\) corresponds to the largest integer that isn't greater than \(x\).
• It produces an output that has practical applications in various disciplines, including computer science and number theory.
• Any deviation above an integer value results in the next whole number being moved to.

When employed in a mathematical equation, such as the given piecewise function \(3[x]\), it creates a stepped pattern by multiplying each integer output by a constant, illustrating its utility in creating predictable, discrete levels.