Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(f(x)=\llbracket 2 x-1 \rrbracket\)

Step-by-Step Solution

Answer

The function \( f(x) = \lfloor 2x - 1 \rfloor \) is not continuous at half-integer x-values. The function jumps from one integer to the next at these points due to the nature of the floor function.

Step 1: Graphing the Function

Use a graphing utility to graph the function \( f(x) = \lfloor 2x - 1 \rfloor \). The steps to do this would depend on the specific utility being used, but in general, this function must be input and the resulting graph observed.

Step 2: Identifying Points of Discontinuity

After plotting the graph, we see that there are vertical jumps at every point where x is a half-integer (e.g., -1.5, -0.5, 0.5, 1.5, etc.). These are the points of discontinuity, where the function jumps from one integer to the next.

Step 3: Explanation of Discontinuity

The function is discontinuous at these points because the floor function rounds down to the nearest integer. As x approaches a half-integer from the left, \(2x - 1\) approaches one integer, and as x moves past the half-integer, \(2x - 1\) quickly jumps to the next integer. Since there's no smooth transition, the function is discontinuous at these points.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize functions. These tools, which can be software or handheld devices, allow users to input mathematical expressions and produce graphical representations. By graphing a function, you can observe its behavior across different values of \( x \). This is particularly helpful for identifying patterns and characteristics, such as points of discontinuity.

• To use a graphing utility, input the function syntax according to the utility's requirements.
• Most graphing utilities offer zoom and trace functions to examine specific parts of the graph closely.
• Using these tools, one can easily spot sections where a function might exhibit unusual behaviors, such as jumps or breaks.

Visualizing the function \( f(x) = \lfloor 2x - 1 \rfloor \) with a graphing utility reveals the nature of the function and helps identify discontinuities.

Floor Function

The floor function, denoted by \( \lfloor x \rfloor \), is a type of step function that rounds a real number down to the nearest integer. This function is important in mathematics, especially in contexts where discrete solutions are necessary.

• The floor function takes any real number and converts it to the largest integer less than or equal to it.
• For example, \( \lfloor 2.7 \rfloor = 2 \) and \( \lfloor -0.3 \rfloor = -1 \).
• When graphed, the floor function appears as a series of flat, horizontal steps, which result in vertical jumps at integer crossings.

Understanding the floor function is key to analyzing any behavior related to step-like patterns, such as the one encountered in the function \( f(x) = \lfloor 2x - 1 \rfloor \).

Points of Discontinuity

Points of discontinuity are specific values where a function does not behave in a consistent, smooth manner. Instead, there are abrupt changes or jumps.

• For the function \( f(x) = \lfloor 2x - 1 \rfloor \), these discontinuities are found at every point where \( x \) is a half-integer, like -1.5, -0.5, 0.5, etc.
• Here, as \( x \) crosses these half-integer values, \( 2x - 1 \) jumps from one integer to the next, creating a break in the graph.
• This type of behavior is typical of floor functions and other similar piecewise functions.

Points of discontinuity are crucial for understanding the limitations and behavior of non-continuous functions.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. They can combine various functions and behaviors within a single framework.

• A piecewise function consists of multiple sub-functions, each applying to specific intervals of the main function’s domain.
• They can include step functions, linear functions, or any other type of mathematical expressions.
• The function \( f(x) = \lfloor 2x - 1 \rfloor \) is an example of a piecewise function because it has different constant values based on ranges of \( x \).

These functions are incredibly useful for modeling situations where a rule changes based on varying conditions. By understanding piecewise functions, one can grasp how a seemingly simple function can exhibit complex behaviors across its domain.