Linear Function G(x) = 2x - 3 Table Range And Graph Discussion

Hey guys! Let's dive into the fascinating world of linear functions, focusing specifically on the function g(x) = 2x - 3. We're going to break down everything you need to know about this function, from creating tables and determining its range to understanding its graphical representation. So, buckle up, and let's get started!

Understanding Linear Functions

Before we jump into the specifics of g(x) = 2x - 3, let's take a step back and discuss what linear functions are all about. At its core, linear functions are mathematical expressions that, when graphed, produce a straight line. These functions follow a general form: f(x) = mx + c, where 'm' represents the slope (the steepness of the line) and 'c' represents the y-intercept (the point where the line crosses the y-axis). The beauty of linear functions lies in their simplicity and predictability. They exhibit a constant rate of change, meaning that for every unit increase in 'x', 'y' changes by a consistent amount (the slope). This consistent behavior makes them incredibly useful in modeling real-world scenarios, from calculating distances traveled at a constant speed to predicting the growth of a plant over time.

Linear functions are pervasive in various fields, demonstrating their practical significance. In physics, they can represent motion at a constant velocity, where the slope indicates the speed and the y-intercept represents the initial position. In economics, they can model cost functions, where the slope represents the variable cost per unit and the y-intercept represents the fixed costs. Even in everyday life, linear functions can help us understand things like the relationship between hours worked and wages earned, where the slope represents the hourly wage and the y-intercept represents any base salary. The ability to represent these diverse phenomena with a simple equation highlights the power and versatility of linear functions. When working with linear functions, it's important to remember the key parameters that define them: the slope and the y-intercept. The slope, often denoted by 'm', quantifies the steepness and direction of the line. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The magnitude of the slope reflects how quickly the line rises or falls; a larger magnitude corresponds to a steeper line. The y-intercept, denoted by 'c', is the point where the line intersects the vertical axis. It represents the value of the function when x is zero, providing a crucial reference point for understanding the function's behavior. By carefully analyzing the slope and y-intercept, we can gain valuable insights into the linear relationship being modeled and make accurate predictions about its behavior.

Delving into g(x) = 2x - 3

Now, let's focus on our specific function: g(x) = 2x - 3. Comparing it to the general form f(x) = mx + c, we can quickly identify that the slope (m) is 2 and the y-intercept (c) is -3. This tells us that the line will be sloping upwards (positive slope) and will cross the y-axis at the point (0, -3). Understanding the slope and y-intercept provides a solid foundation for analyzing the function's behavior. The slope of 2 indicates that for every increase of 1 in the value of 'x', the value of 'g(x)' increases by 2. This consistent rate of change is a hallmark of linear functions, making them easy to predict and understand. The y-intercept of -3 tells us that when 'x' is 0, the function's value is -3. This point serves as a starting point for visualizing the line and understanding its position on the coordinate plane. To further explore this linear function, we can also consider its x-intercept, which is the point where the line crosses the x-axis. To find the x-intercept, we set g(x) equal to 0 and solve for x: 0 = 2x - 3. Adding 3 to both sides gives us 3 = 2x, and dividing by 2 yields x = 1.5. Therefore, the x-intercept is the point (1.5, 0). Knowing both the x and y-intercepts provides valuable anchors for graphing the line accurately. With a firm grasp of the slope, y-intercept, and x-intercept, we can confidently analyze and visualize the behavior of the linear function g(x) = 2x - 3.

Creating a Table of Values

To get a better feel for how g(x) = 2x - 3 behaves, let's create a table of values. This involves choosing several values for 'x' and then calculating the corresponding values for g(x). This is a super helpful way to visualize the relationship between the input ('x') and the output (g(x)). By plugging in different values for 'x', we can observe how the function transforms these inputs into outputs, revealing the linear pattern. The choice of 'x' values is crucial for creating a representative table. Typically, we select a range of values, including both positive and negative numbers, as well as zero, to capture the full spectrum of the function's behavior. Simple values like -2, -1, 0, 1, and 2 are often good starting points. However, depending on the specific function and the level of detail desired, we might choose a wider range or more closely spaced values. Once the 'x' values are selected, we substitute each one into the function's equation, g(x) = 2x - 3, and perform the calculation to obtain the corresponding g(x) value. For instance, when x = -2, g(-2) = 2(-2) - 3 = -4 - 3 = -7. Similarly, when x = 0, g(0) = 2(0) - 3 = -3. These calculations provide us with ordered pairs (x, g(x)) that represent points on the line. By plotting these points on a coordinate plane, we can begin to visualize the linear relationship and confirm our understanding of the function's behavior.

Here’s a small table as an example:

Determining the Range

The range of a function is the set of all possible output values (g(x) in this case). For linear functions, unless there are specific restrictions, the range is usually all real numbers. Why? Because the line extends infinitely in both directions! The range of a function is a fundamental concept in mathematics, representing the set of all possible output values that the function can produce. It provides a comprehensive view of the function's behavior and its ability to map inputs to outputs. For linear functions, the range is typically all real numbers, denoted as (-∞, ∞), unless there are specific constraints or limitations imposed on the function's domain or definition. The reason for this expansive range lies in the nature of linear functions themselves. Linear functions, characterized by their constant rate of change, extend infinitely in both directions, both horizontally and vertically. This means that as the input 'x' varies across the real number line, the output g(x) will also traverse the entire real number line, without any gaps or interruptions. The slope of the linear function, which determines its steepness and direction, plays a crucial role in this unbounded range. A non-zero slope ensures that the line will continue to rise or fall indefinitely, covering all possible y-values. Only in the special case of a horizontal line, where the slope is zero, will the range be restricted to a single value, the y-intercept.

Graphing g(x) = 2x - 3

Now for the fun part: graphing! To graph g(x) = 2x - 3, we can use the information we've already gathered. We know the y-intercept is (0, -3), and we can use the slope (2) to find other points. Remember, a slope of 2 means