Problem 14 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=[x+3]\)

Step-by-Step Solution

Answer

Domain: \( \mathbb{R} \), Range: integers. The graph shows stepwise increments for each unit increase in \( x \).

Step 1: Understand the Function

The function given is \( f(x) = [x+3] \), which is the greatest integer function or floor function. This means for any input \( x \), the function outputs the greatest integer less than or equal to \( x+3 \). This creates a step-like graph.

Step 2: Determine Key Points

Since the function is \( f(x) = [x+3] \), we examine key integer inputs to understand the pattern of the function. For example:- If \( x = -3 \), then \( f(x) = [-3+3] = [0] = 0 \)- If \( x = -2.5 \), then \( f(x) = [-2.5+3] = [0.5] = 0 \)- If \( x = -2 \), then \( f(x) = [-2+3] = [1] = 1 \)Compute a few more points in a similar way to determine the pattern.

Step 3: Sketch the Graph

Plot the determined points from Step 2 on a coordinate graph. The output is constant between integers, resulting in horizontal steps. For example, the function remains at 0 from \( x = -3 \) to \( x = -2 \), then jumps to 1 at \( x = -2 \) and remains until \( x = -1 \).

Step 4: Identify the Domain

Since \( f(x) = [x+3] \) is defined for all real numbers, the domain of the function is all real numbers \( \mathbb{R} \). A function using the greatest integer operator is unbounded in the horizontal direction.

Step 5: Identify the Range

The range of \( f(x) = [x+3] \) consists only of integer values. This is because the output of the greatest integer function is always an integer.

Key Concepts

Greatest Integer Function

The greatest integer function, often notated as \( \lfloor x \rfloor \), outputs the largest integer that is less than or equal to a given input. It is also known as the floor function. For example, if the input value is 3.7, the greatest integer function will output 3, since 3 is the largest integer not greater than 3.7. Similarly, for an input of -1.2, the output will be -2, as -2 is the largest integer less than -1.2.

This function creates a characteristic 'step' pattern when graphed, hence its nickname as the "step function." At each integer point, the value of the function increases by one, creating a visible step in a graph. When solving problems with greatest integer functions, it's important to recognize this step nature, as it impacts both the graph appearance and the interpretation of results.

Domain and Range

In graphing functions, understanding the domain and range is crucial.

The **domain** of a function refers to all possible input values. For the greatest integer function \( f(x) = [x+3] \), the domain is all real numbers \( \mathbb{R} \). This means that any real number can be input into the function, and it will return a valid integer output.

• The domain of \( f(x) = [x+3] \) is \( (-\infty, \infty) \) or \( \mathbb{R} \).

The **range** of a function consists of all possible output values. For this function, the range consists solely of integer values. Regardless of the precise input, the greatest integer function always outputs an integer, which is why the range is a set of integers.

• The range of \( f(x) = [x+3] \) is the set of all integers \( \mathbb{Z} \).

Floor Function

The floor function is another name for the greatest integer function. It is essential in discrete mathematics and computer science. As its name suggests, it 'floors' a real input to the nearest lower integer. The floor function is denoted by \( \lfloor x \rfloor \).

For instance:

• \( \lfloor 3.5 \rfloor = 3 \)
• \( \lfloor -2.1 \rfloor = -3 \)
• \( \lfloor 7 \rfloor = 7 \)

When applying the floor function in graphing, you observe that the output leaps at each integer input point. Outside of graphing scenarios, the floor function finds utility in rounding down operations, ensuring programming sequences rely on whole numbers, and discretizing continuous data sets.

Keep in mind, the behavior of this function is consistent, making it predictable and reliable for these types of applications.

Step Function

The term "step function" is often used to describe the visual appearance of the graph of the greatest integer function.
As mentioned, this function exhibits a series of flat, horizontal lines when graphed—akin to steps on a staircase.

Each "step" corresponds to a range of initial input values that result in the same output integer.

• For example, in \( f(x) = [x+3] \), the step at 0 extends from inputs \( x = -3 \) to just below \( x = -2 \).
• The next step at 1 covers from input \( x = -2 \) to just below \( x = -1 \).

This makes understanding the step function vital for sketching and analyzing piecewise linear graphs. In real-world applications, step functions can represent situations where a quantity changes abruptly rather than gradually, such as tax brackets or pricing tiers. Knowing how to work with step functions hence applies beyond just mathematical theory, also encompassing practical, everyday scenarios.