Exploring The Function G(x) = 2x – 3 A Comprehensive Guide
Hey guys! Today, we're diving deep into the world of functions, specifically the function g(x) = 2x – 3. We're going to break it down step by step, creating a mapping table, figuring out the range, and even plotting its graph on a Cartesian coordinate system. So, buckle up and let's get started!
Understanding the Function g(x) = 2x – 3
Before we jump into the specifics, let's first understand what this function actually means. The function g(x) = 2x – 3 is a linear function, which means it represents a straight line when graphed. The 'x' is our input, and 'g(x)' is the output we get after applying the function's rule. In this case, the rule is: take the input 'x', multiply it by 2, and then subtract 3. The domain {x | -2 ≤ x ≤ 2, x ∈ R} tells us the allowed values for 'x'. It means 'x' can be any real number between -2 and 2, including -2 and 2.
To truly grasp the essence of this function, let's delve deeper into its components and behavior. The function g(x) = 2x - 3 is a linear function, a fundamental concept in algebra. Linear functions are characterized by their constant rate of change, which in this case is represented by the coefficient of x, which is 2. This means that for every unit increase in x, the value of g(x) increases by 2. The constant term, -3, represents the y-intercept, the point where the line crosses the y-axis. Understanding these fundamental aspects of the function is key to predicting its behavior and interpreting its graph. We'll see how these concepts come into play as we construct the mapping table and plot the graph. The domain, as mentioned earlier, plays a crucial role in defining the function's scope. It limits the set of possible input values, which in turn affects the output values and the overall shape of the graph within the specified interval. Considering the domain helps us focus our analysis and provides a clear boundary for our investigation. Without a defined domain, the function could extend infinitely in both directions, making it challenging to analyze its specific behavior within a given range. So, let's keep these important aspects in mind as we move forward and explore the different facets of the function g(x) = 2x - 3. Remember, a solid understanding of the basics is the foundation for more complex mathematical concepts, and this function provides a great starting point for our journey!
Creating the Mapping Table
Our first task is to create a mapping table. This table will show us how different 'x' values within our domain are transformed into 'g(x)' values. To do this, we'll select a few 'x' values within the domain -2 ≤ x ≤ 2. Let's choose some easy-to-work-with values like -2, -1, 0, 1, and 2. Now, we'll plug each of these 'x' values into our function g(x) = 2x – 3 and calculate the corresponding 'g(x)' values.
Let's break down the process of creating a mapping table for the function g(x) = 2x - 3 in more detail. This table will serve as a visual representation of the function's behavior, showing the relationship between input (x) and output (g(x)) values. As we mentioned, selecting appropriate 'x' values within the domain is crucial for constructing an accurate and informative table. Choosing a diverse set of values, including both negative and positive numbers, as well as zero, helps us capture the function's behavior across the entire domain. In our case, we've chosen -2, -1, 0, 1, and 2, which provide a good spread within the domain -2 ≤ x ≤ 2. Now, let's delve into the actual calculations. For each 'x' value, we substitute it into the function g(x) = 2x - 3 and perform the arithmetic operations. For instance, when x = -2, we have g(-2) = 2(-2) - 3 = -4 - 3 = -7. Similarly, for x = -1, we have g(-1) = 2(-1) - 3 = -2 - 3 = -5. We repeat this process for all the chosen 'x' values, carefully calculating each corresponding 'g(x)' value. Accuracy is paramount in this step, as any error in calculation will lead to an incorrect mapping table and ultimately affect our understanding of the function. Once we have calculated all the 'g(x)' values, we can organize them into a table format, with 'x' values in one column and their corresponding 'g(x)' values in another. This table becomes a valuable tool for visualizing the function's behavior and identifying patterns. It also serves as a foundation for graphing the function, as we'll see in the next section. So, let's proceed with the calculations and construct our mapping table, ensuring we're meticulous in our approach to achieve an accurate representation of the function g(x) = 2x - 3.
- For x = -2: g(-2) = 2(-2) – 3 = -4 – 3 = -7
- For x = -1: g(-1) = 2(-1) – 3 = -2 – 3 = -5
- For x = 0: g(0) = 2(0) – 3 = 0 – 3 = -3
- For x = 1: g(1) = 2(1) – 3 = 2 – 3 = -1
- For x = 2: g(2) = 2(2) – 3 = 4 – 3 = 1
Now, we can put these results into a table:
| x | g(x) |
|---|---|
| -2 | -7 |
| -1 | -5 |
| 0 | -3 |
| 1 | -1 |
| 2 | 1 |
Determining the Range of the Function
The range of a function is the set of all possible output values (g(x) in this case) that we can get when we plug in the 'x' values from the domain. Looking at our mapping table, we can see the g(x) values are -7, -5, -3, -1, and 1. Since our domain includes all real numbers between -2 and 2, the range will also include all real numbers between the smallest and largest g(x) values we found. Therefore, the range of the function is -7 ≤ g(x) ≤ 1.
To fully understand how we determine the range of the function g(x) = 2x - 3, let's delve into the underlying principles and consider the function's characteristics. The range, as we mentioned, represents the set of all possible output values (g(x)) that the function can produce within the given domain. In our case, the domain is -2 ≤ x ≤ 2, meaning 'x' can take on any real value between -2 and 2, inclusive. Since g(x) = 2x - 3 is a linear function, it exhibits a continuous and consistent behavior. This means that as 'x' varies continuously within the domain, g(x) will also vary continuously, without any breaks or jumps. To find the range, we need to identify the minimum and maximum values of g(x) within the domain. Because the function is linear and has a positive slope (the coefficient of x is 2), the minimum value of g(x) will occur at the minimum value of x, which is -2. Similarly, the maximum value of g(x) will occur at the maximum value of x, which is 2. We've already calculated g(-2) = -7 and g(2) = 1 in our mapping table. These values represent the lower and upper bounds of the range, respectively. Since the function is continuous, it will take on all values between -7 and 1 as 'x' varies between -2 and 2. Therefore, the range of the function g(x) = 2x - 3 within the domain -2 ≤ x ≤ 2 is -7 ≤ g(x) ≤ 1. This means that the output values of the function will always fall within this interval, providing a clear understanding of the function's behavior and limitations. Understanding the concept of range is crucial for analyzing and interpreting functions, as it helps us determine the possible output values and their distribution within a given domain. So, by carefully considering the function's characteristics and the domain's boundaries, we can accurately determine the range and gain valuable insights into the function's overall behavior.
Graphing the Function on the Cartesian Coordinate System
Now, let's visualize our function by graphing it on the Cartesian coordinate system. The Cartesian coordinate system, also known as the x-y plane, is a two-dimensional space defined by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point in this plane is represented by an ordered pair (x, y), where 'x' is the point's horizontal position and 'y' is its vertical position.
To graph our function g(x) = 2x – 3, we'll use the points we found in our mapping table. Each (x, g(x)) pair represents a point on the graph. So, we have the points (-2, -7), (-1, -5), (0, -3), (1, -1), and (2, 1). We'll plot each of these points on the coordinate plane. Start by finding the x-coordinate on the x-axis, then move vertically until you reach the corresponding g(x) value (the y-coordinate) on the y-axis. Mark the point there.
Once we've plotted all the points, we'll notice they form a straight line. This is because g(x) = 2x – 3 is a linear function. To complete the graph, we'll draw a straight line through these points. Remember, our domain is -2 ≤ x ≤ 2, so our line will only extend between x = -2 and x = 2. This means we'll have a line segment rather than an infinite line.
Let's delve deeper into the process of graphing the function g(x) = 2x - 3 on the Cartesian coordinate system. This visual representation will provide a clear understanding of the function's behavior and its relationship between input and output values. As we discussed, the Cartesian coordinate system is the foundation for graphing functions. It allows us to plot points and visualize relationships in a two-dimensional space. The x-axis represents the input values (x), and the y-axis represents the output values (g(x)). To graph our function, we utilize the points we generated in the mapping table. Each point (x, g(x)) represents a specific location on the coordinate plane. For example, the point (-2, -7) indicates that when x is -2, g(x) is -7. To plot this point, we locate -2 on the x-axis and then move vertically downwards until we reach -7 on the y-axis. We mark this location with a dot. We repeat this process for all the points in our mapping table: (-1, -5), (0, -3), (1, -1), and (2, 1). As we plot these points, we'll notice a distinct pattern: they align in a straight line. This is a key characteristic of linear functions. The straight line visually confirms the constant rate of change inherent in the function g(x) = 2x - 3. Now, to complete the graph, we connect the plotted points with a straight line. This line represents all the possible (x, g(x)) pairs within the function's domain. However, it's crucial to remember that our domain is restricted to -2 ≤ x ≤ 2. This means that our graph will be a line segment, not an infinite line. The line segment will start at the point (-2, -7) and end at the point (2, 1), representing the function's behavior within the specified domain. The resulting graph provides a powerful visual representation of the function g(x) = 2x - 3. It clearly shows the linear relationship between x and g(x), the function's slope, and its y-intercept. By understanding how to graph functions on the Cartesian coordinate system, we can gain valuable insights into their behavior and properties. So, let's take a moment to visualize the graph of g(x) = 2x - 3, appreciating the connection between the algebraic equation and its geometric representation.
This line segment visually represents the function g(x) = 2x – 3 within the given domain. You'll see how the line slopes upwards, reflecting the positive coefficient (2) in our function. The line intersects the y-axis at -3, which is our y-intercept.
Conclusion
And there you have it! We've successfully explored the function g(x) = 2x – 3. We created a mapping table, determined the range, and graphed the function on the Cartesian coordinate system. This exercise demonstrates how we can analyze and understand functions by breaking them down into smaller, manageable steps. Remember, math can be fun, especially when you understand the concepts. Keep practicing, and you'll become a function whiz in no time!
