Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=\llbracket x-1 \rrbracket\)

Step-by-Step Solution

Answer

The graph of \(f(x)=\llbracket x-1 \rrbracket\) is a step function shifted to the right by 1 unit compared to the parent function \(f(x) = \llbracket x \rrbracket\).

Step 1: Understand floor function properties

The graph of a floor function is a step function and changes at every integer. For \(f(x) = \llbracket x \rrbracket\), it is understood that the graph is a series of horizontal lines with open and closed intervals.

Step 2: Graph the parent function

Start by graphing the parent function, \(f(x) = \llbracket x \rrbracket\). It will start from the origin (0,0), move to (1,0), (1,1) and so on. This will create a step like structure

Step 3: Graph the given function

After understanding the parent function, it's easier to graph \(f(x) = \llbracket x-1 \rrbracket\). The 'x-1' inside the bracket means the graph will be moved to right by 1. Shift each point of the original function one step to the right. It will start from (1,0), move to (2,0), (2,1) and so on. The desired graph of \(f(x) = \llbracket x-1 \rrbracket\) is obtained.

Key Concepts

Graphing Step Functions

A step function is a type of piecewise function characterized by its discontinuous nature, where the function output remains constant over an interval, then jumps to a new constant level. This type of graph looks like a staircase. Here's some main points to remember when you're dealing with step functions:

• The floor function, expressed as \( f(x) = \lfloor x \rfloor \), returns the greatest integer less than or equal to \( x \).
• The graph is composed of horizontal line segments. Each segment represents the interval from one integer to the next.
• At integer points, the graph "steps" to the next value, hence the name "step function." The left endpoint of each interval is closed (solid dot), and the right endpoint is open (hollow dot), indicating the boundaries of inclusion.

To visualize this, start at the origin (0,0) and draw a horizontal line to (1,0). Then jump to (1,1) and draw another horizontal line to (2,1), and continue this pattern to create the steps.

Horizontal Shifts in Functions

Horizontal shifts occur when the input variable \( x \) is adjusted before being subjected to the function. With floor functions like \( f(x) = \lfloor x-1 \rfloor \), this shift is straightforward to visualize because it affects where the steps start and end.

• In general, the expression \( f(x) = \lfloor x - a \rfloor \) will shift the graph of \( \lfloor x \rfloor \) to the right by \( a \) units.
• If \( a \) is positive, like in our example \( \lfloor x-1 \rfloor \), each step in the graph slides one unit to the right.
• If \( a \) is negative, the graph shifts to the left.

When graphing the function \( f(x) = \lfloor x-1 \rfloor \), take the original step graph \( \lfloor x \rfloor \) and move every point one unit toward the right. Therefore, the step starting at the origin now begins at (1,0).

Piecewise Functions

Piecewise functions are functions composed of multiple sub-functions, each defined over a certain interval. Each sub-function applies to a particular "piece" of the domain, making it versatile for describing situations with different behaviors in different ranges.

• A common example of a piecewise function is our step function, where the floor function creates intervals with fixed outputs.
• The definition of piecewise functions involves specifying how the function behaves on contiguous segments of the x-axis.
• Visually, the graph of a piecewise function can look quite different in each segment, creating a segmented appearance.

For a step function like \( \lfloor x \rfloor \), think of each step as a separate piece defined by the following sub-function: On any given interval \( [n, n+1) \), \( f(x) = n \). These segments combine to form a comprehensive graph that appears as a series of steps.