Question

Describe what is meant by a piece wise linear function.

Short answer

A piecewise linear function is a continuous function defined by a collection of linear functions on separate intervals that are pieced together to form a continuous function. It is composed of multiple linear functions, each defined on a specific interval or subdomain. Some of its properties are: 1. The function is continuous, meaning there are no gaps or breaks in the graph. 2. Each segment of the function is a linear function, meaning it has a constant slope and is represented by a straight line. 3. The function may have multiple subdomains, with each linear function being defined on a specific interval.

Step-by-step solution

Definition

A piecewise linear function is a function that is defined by a collection of linear functions on separate intervals that are pieced together to form a continuous function. In other words, a piecewise linear function is composed of multiple linear functions, each defined on a specific interval or subdomain.

Properties

Some properties of a piecewise linear function are: 1. The function is continuous, meaning there are no gaps or breaks in the graph. 2. Each segment of the function is a linear function, meaning it has a constant slope and is represented by a straight line. 3. The function may have multiple subdomains, with each linear function being defined on a specific interval.

Example

Let's consider an example of a piecewise linear function: \[ f(x) = \begin{cases} x, & 0 \le x < 2 \\ -x+4, & 2 \le x \le 4 \end{cases} \] This function has two linear functions, \(f_1(x)=x\) and \(f_2(x)=-x+4\), each defined on separate intervals. \(f_1(x)\) is defined on the interval \([0,2)\) and \(f_2(x)\) is defined on the interval \([2,4]\). The function is continuous because the two linear functions connect at the point \((2,2)\), so the graph of this piecewise linear function doesn't have any breaks or gaps.

Key concepts

Continuous Function

A continuous function is a function whose graph can be drawn without lifting your pencil from the paper. It means there are no breaks or gaps along its path.
When we talk about piecewise linear functions, continuity becomes an essential part because these functions are made of different linear pieces that are linked together.
In mathematical terms, a function is continuous at a point if the left-hand and right-hand limits as the input approaches the point are equal to the function's value at that point.
For example, in a piecewise function made of segments like \( f(x) = x \) for \( 0 \leq x < 2 \) and \( f(x) = -x + 4 \) for \( 2 \leq x \leq 4 \), continuity means the value matches at the connecting point.
An easy way to check continuity in piecewise functions is to ensure that the endpoints of each piece align properly with the start of the next, creating a seamless graph.

Linear Functions

Linear functions are the simplest type of functions to understand. They represent relationships between two variables with a constant rate of change or slope.
The general form of a linear function is \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.
Each straight line portrays how one variable changes in response to another linearly.
For piecewise linear functions, each segment within the piecewise function is a linear function.
Consider the example \( f_1(x) = x \) and \( f_2(x) = -x + 4 \); both are linear functions with respective slopes of 1 and -1.
These functions can describe real-world situations where different phases of a process follow distinct linear trends, linked together smoothly to form a comprehensive description.

Function Intervals

Function intervals are specific ranges along the x-axis where particular parts of a function are defined.
For piecewise linear functions, these intervals determine where each linear component applies.

• For instance, one portion of the function might be active between \( x = 0 \) and \( x = 2 \), while another takes over from \( x = 2 \) to \( x = 4 \).

Each interval in the piecewise function can potentially highlight a different rule or behavior for the function.
Understanding intervals is crucial because it tells us where the function switches from one linear equation to another.
It helps in accurately plotting or visualizing the function.
The intervals must be specified clearly to ensure the function's entire path and behavior are understood precisely.
For example, in our example function, the transition happens smoothly at \( x = 2 \) from \( f_1(x) \) to \( f_2(x) \).
This ensures that you see the whole picture of how the piecewise function behaves across different domains.