Graphing And Range Of Piecewise Function F(x) = {2x If X ≤ 0, 2 If X > 0}

This article explores the piecewise function ${ f(x) }$ defined as:

${ f(x)=\left\{\begin{array}{rll} 2 x & \text { if } & x \leq 0 \\ 2 & \text { if } & x > 0 \end{array}\right. }$

We will graph this function and then use the graph to determine its range. Understanding piecewise functions is crucial in various areas of mathematics, as they allow us to model situations where the function's behavior changes based on the input value. This article aims to provide a comprehensive guide on how to analyze and graph piecewise functions, ensuring a solid understanding of their properties and applications.

1. Graphing the Piecewise Function

To effectively graph this piecewise function, we need to consider each piece separately over its specified interval. This meticulous approach ensures that we accurately represent the function's behavior across its entire domain.

Understanding the First Piece: ${ 2x }$ for ${ x \leq 0 }$

Let’s begin by examining the first piece of the function, which is ${ f(x) = 2x }$ when ${ x \leq 0 }$. This part of the function represents a straight line with a slope of 2. The slope indicates how steeply the line rises or falls, and in this case, for every unit increase in ${ x }$, the value of ${ f(x) }$ increases by 2. The line also passes through the origin (0, 0) because when ${ x = 0 }$, ${ f(0) = 2 \times 0 = 0 }$. This point serves as an anchor for our graph, helping us to accurately position the line on the coordinate plane.

Since this piece is defined for ${ x \leq 0 }$, we only consider the part of the line that lies on the left side of the y-axis and includes the y-axis itself. To graph this, we can plot a few points. For instance, when ${ x = -1 }$, ${ f(-1) = 2 \times -1 = -2 }$, giving us the point (-1, -2). Similarly, when ${ x = -2 }$, ${ f(-2) = 2 \times -2 = -4 }$, resulting in the point (-2, -4). By connecting these points, we can draw the line segment for this part of the function. It's important to note that the endpoint at ${ x = 0 }$ is included because the inequality is ${ x \leq 0 }$, so we use a closed circle (or a solid dot) at the origin to indicate this inclusion.

Analyzing the Second Piece: ${ 2 }$ for ${ x > 0 }$

The second piece of the function is ${ f(x) = 2 }$ when ${ x > 0 }$. This is a horizontal line at ${ y = 2 }$. A horizontal line signifies that the function's value remains constant regardless of the ${ x }$ value. In this case, for any ${ x }$ greater than 0, the function's value is always 2. This constant behavior is a key characteristic of horizontal lines and is straightforward to graph.

Since this piece is defined for ${ x > 0 }$, we only consider the part of the line that lies to the right of the y-axis. To graph this, we draw a horizontal line at ${ y = 2 }$. However, since the inequality is strict (${ x > 0 }$), the endpoint at ${ x = 0 }$ is not included. We represent this exclusion by using an open circle at the point (0, 2). This open circle visually indicates that the function approaches this point but does not actually include it.

Combining the Pieces on the Graph

To complete the graph of the piecewise function, we combine the graphs of both pieces on the same coordinate plane. On the left side of the y-axis (including the y-axis), we have the line segment representing ${ f(x) = 2x }$. On the right side of the y-axis, we have the horizontal line at ${ y = 2 }$, with an open circle at (0, 2). This combination gives us a clear picture of the function's behavior across its entire domain. The graph visually demonstrates how the function's definition changes at ${ x = 0 }$, transitioning from a linear function to a constant function. This graphical representation is invaluable for understanding the function's overall characteristics and behavior.

By plotting these two pieces together, we get the complete graph of the piecewise function. This graph clearly shows the function's behavior as it transitions from a linear segment to a constant value at ${ x = 0 }$.

2. Determining the Function's Range

After graphing the piecewise function, the next crucial step is to determine its range. The range of a function is the set of all possible output values (y-values) that the function can produce. Identifying the range helps us understand the function's vertical extent and the set of values it can attain. We analyze the graph to find the function's range, looking for the lowest and highest y-values that the function reaches. This visual approach, combined with an understanding of the function's definition, allows us to accurately determine the range.

Analyzing the Graph for Output Values

To determine the range, we need to examine the y-values covered by the graph. The first piece, ${ f(x) = 2x }$ for ${ x \leq 0 }$, is a line that extends from the origin (0, 0) downwards to negative infinity. This means that the function takes on all y-values less than or equal to 0 for this piece. The line's slope of 2 ensures that as ${ x }$ decreases, ${ f(x) }$ also decreases without bound, covering the entire negative y-axis.

The second piece, ${ f(x) = 2 }$ for ${ x > 0 }$, is a horizontal line at ${ y = 2 }$. This part of the function only has one y-value, which is 2. However, because the inequality is ${ x > 0 }$, the point (0, 2) is not included (indicated by the open circle on the graph). Despite this exclusion, the y-value of 2 is still part of the function's range, as the function's value is exactly 2 for all ${ x }$ greater than 0. This constant value contributes a single, specific element to the overall range.

Combining the Output Values to Define the Range

Now, we combine the y-values from both pieces to find the overall range of the piecewise function. The first piece contributes all y-values less than or equal to 0, which can be represented as the interval ${ (-\infty, 0] }$. This interval includes all real numbers from negative infinity up to and including 0. The second piece contributes the single y-value of 2. Combining these, the range of the function includes all numbers in the interval ${ (-\infty, 0] }$ and the single value 2. This means that the function can output any value that is non-positive, as well as the value 2.

Expressing the Range in Interval Notation

The range of the function can be expressed in interval notation as the union of the interval ${ (-\infty, 0] }$ and the set containing the single value 2. Mathematically, this is written as:

${ (-\infty, 0] \cup \{2\} }$

This notation clearly indicates that the range includes all y-values from negative infinity up to and including 0, as well as the specific y-value of 2. The use of interval notation provides a concise and precise way to represent the range of the function, making it easy to communicate and interpret the function's output values.

Therefore, the range of the piecewise function is ${ (-\infty, 0] \cup \{2\} }$. This means the function can output any value less than or equal to 0, as well as the value 2.

Conclusion

In summary, we successfully graphed the given piecewise function and determined its range by carefully analyzing each piece and its corresponding interval. The graph provides a visual representation of the function's behavior, while the range ${ (-\infty, 0] \cup \{2\} }$ accurately describes the set of all possible output values. Understanding how to graph and determine the range of piecewise functions is a valuable skill in mathematics, with applications in various fields where functions change behavior based on input conditions.