Finding The X-intercept Of G(x) = 3x - 6 A Step-by-Step Guide

by ADMIN 62 views

In mathematics, determining the x-intercept of a function is a fundamental concept, especially when analyzing graphs and solving equations. The x-intercept represents the point where the graph of the function intersects the x-axis. At this point, the y-coordinate (or the function's value) is zero. This article provides a comprehensive guide on how to find the x-intercept of the function g(x) = 3x - 6, a linear function, and explains the underlying principles behind this process. This concept is crucial not only in algebra but also in more advanced mathematical fields such as calculus and real analysis, as it lays the groundwork for understanding roots, zeros, and solutions of equations. We will delve into the algebraic methods used to solve this problem, and also illustrate how this concept is visually represented on a graph. Moreover, understanding x-intercepts is not just a theoretical exercise; it has practical applications in various fields, including physics, engineering, and economics, where functions are used to model real-world phenomena. For instance, in economics, the x-intercept of a supply or demand curve can represent the point at which there is no supply or demand in the market. In physics, it could represent the time at which an object's displacement is zero. Therefore, mastering the skill of finding x-intercepts is essential for anyone seeking to apply mathematical concepts to real-world problems. Furthermore, we will discuss common pitfalls and misconceptions that students often encounter when dealing with x-intercepts, ensuring a thorough understanding of the topic. By the end of this article, you should be able to confidently identify and calculate the x-intercepts of linear functions and understand their significance in both theoretical and practical contexts. This understanding will not only help you in your academic pursuits but also equip you with a valuable tool for problem-solving in various aspects of life.

Understanding x-intercepts

The x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept of a function, we set the function equal to zero and solve for x. In the context of the given function, g(x) = 3x - 6, this means we need to solve the equation 3x - 6 = 0. This concept is foundational in coordinate geometry and is crucial for understanding the behavior of functions. The x-intercept not only provides a specific point on the graph but also gives valuable information about the function's roots or solutions. In many real-world scenarios, the x-intercept holds significant meaning. For example, in a business context, if a function represents the profit of a company, the x-intercept could represent the break-even point, where the company neither makes a profit nor incurs a loss. Similarly, in physics, if a function represents the height of a projectile over time, the x-intercept could represent the time at which the projectile hits the ground. Understanding x-intercepts also helps in sketching the graph of a function. Knowing the x-intercept, along with other key points such as the y-intercept and the slope (in the case of linear functions), allows for a more accurate representation of the function's behavior. This is particularly useful in fields like engineering and data analysis, where visual representations of data are essential for interpretation and decision-making. Moreover, the concept of x-intercepts extends beyond simple linear functions. In more complex functions, such as polynomials, finding x-intercepts can involve more sophisticated techniques, including factoring, using the quadratic formula, or employing numerical methods. The number of x-intercepts a function has can also provide insights into the nature of the function itself. For instance, a quadratic function can have zero, one, or two x-intercepts, depending on its discriminant. Therefore, a solid understanding of x-intercepts is not only essential for basic algebra but also serves as a stepping stone for more advanced mathematical concepts.

Solving for the x-intercept of g(x) = 3x - 6

To find the x-intercept of the function g(x) = 3x - 6, we set g(x) to zero and solve for x. This is because, at the x-intercept, the y-value, which is g(x), is zero. The equation becomes 3x - 6 = 0. To solve this equation, we first add 6 to both sides, resulting in 3x = 6. Then, we divide both sides by 3 to isolate x, giving us x = 2. Therefore, the x-intercept occurs when x is 2. This algebraic process is a fundamental skill in algebra and is used extensively in various mathematical contexts. It’s not just about finding the solution; it’s about understanding the relationship between the equation and the graph it represents. The solution x = 2 tells us that the graph of the function g(x) = 3x - 6 intersects the x-axis at the point where x is 2. This point is crucial for sketching the graph of the function and understanding its behavior. Furthermore, the process of solving for x-intercepts is not limited to linear functions. It can be applied to any type of function, although the methods for solving the equation may vary depending on the function's complexity. For example, finding the x-intercepts of a quadratic function might involve factoring, completing the square, or using the quadratic formula. Similarly, finding the x-intercepts of trigonometric functions might involve understanding the periodic nature of these functions and their values at specific angles. In the context of real-world applications, solving for x-intercepts can provide valuable insights. In a supply-demand model, the x-intercepts can represent points of equilibrium. In a projectile motion problem, the x-intercept can represent the time at which the projectile lands. Therefore, mastering the algebraic techniques for finding x-intercepts is essential for both academic and practical problem-solving. This skill is a building block for more advanced topics in mathematics and its applications in various fields.

Identifying the Correct Option

Now that we've determined the x-intercept occurs at x = 2, we can identify the correct option among the choices provided. The x-intercept is a point on the graph where y = g(x) = 0. We found that this occurs when x = 2. Therefore, the x-intercept is the point (2, 0). Looking at the given options, we need to find the one that represents this point. Option D, (2, g(2)), is the correct choice because when x = 2, g(2) = 3(2) - 6 = 0. Thus, the point is (2, 0), which is the x-intercept. This step of identifying the correct option is crucial as it tests the understanding of the concept of x-intercepts and the ability to apply the solution to the given context. It's not just about finding the value of x; it's about understanding what that value represents in terms of the graph of the function. In this case, the x-intercept is a specific point on the graph, and the correct option must reflect this understanding. The other options can be eliminated because they do not represent the x-intercept. For example, option A, (-3, g(-3)), represents a point where x = -3, which is not the x-intercept we found. Similarly, option B, (0, g(0)), represents the y-intercept, not the x-intercept. Option C, (2, g(0)), is incorrect because g(0) is not 0. Therefore, by carefully analyzing the options and applying our understanding of x-intercepts, we can confidently identify the correct answer. This process highlights the importance of not just solving the problem but also understanding the context and the meaning of the solution. This approach is valuable in mathematics and in problem-solving in general, as it encourages a deeper understanding of the underlying concepts.

Why other options are incorrect

Understanding why the other options are incorrect is as crucial as knowing the correct answer. It solidifies your grasp of the concept and helps avoid similar mistakes in the future. Let's analyze each incorrect option:

  • Option A: (-3, g(-3))

    This option represents a point on the graph where x = -3. To determine if this is the x-intercept, we would need to check if g(-3) = 0. Calculating g(-3), we get g(-3) = 3(-3) - 6 = -9 - 6 = -15, which is not zero. Therefore, this point is not on the x-axis and is not the x-intercept. This option highlights a common misconception: simply choosing a point on the graph does not mean it is the x-intercept. The y-coordinate must be zero for a point to lie on the x-axis. This understanding is crucial for accurately identifying x-intercepts and avoiding this type of error in future problems. Additionally, this analysis underscores the importance of substituting the x-value into the function to verify if the y-value is indeed zero. Without this verification, one might mistakenly identify a point that lies on the graph but not on the x-axis, leading to an incorrect answer. This detailed examination of option A reinforces the fundamental definition of an x-intercept and the process of verifying whether a given point satisfies this definition.

  • Option B: (0, g(0))

    This option represents the y-intercept, not the x-intercept. The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. While this is a significant point on the graph, it is not what we are looking for when asked for the x-intercept. To confirm this, we can calculate g(0): g(0) = 3(0) - 6 = -6. So, the point is (0, -6), which clearly lies on the y-axis. This distinction between x-intercepts and y-intercepts is fundamental in coordinate geometry. Mistaking one for the other is a common error, especially when students are first introduced to these concepts. Therefore, it's essential to understand that the x-intercept is where the graph crosses the x-axis (y = 0), while the y-intercept is where the graph crosses the y-axis (x = 0). This option serves as a good reminder of the importance of carefully reading the question and understanding what is being asked. If the question specifically asks for the x-intercept, one should focus on finding the point where y = 0, not x = 0. This careful attention to detail is a valuable skill in mathematics and problem-solving in general.

  • Option C: (2, g(0))

    This option is tricky because it has the correct x-coordinate (x = 2), which we found to be the x-coordinate of the x-intercept. However, the y-coordinate is g(0), which we already calculated as -6. So, this point is (2, -6), which is not on the x-axis. This option highlights the importance of checking both the x and y coordinates. Even if the x-coordinate is correct, the point will not be the x-intercept unless the y-coordinate is zero. This type of option is often used to test whether students fully understand the definition of an x-intercept or if they are simply memorizing steps without understanding the underlying concepts. The fact that the x-coordinate is correct might lead some students to mistakenly choose this option, but a careful check of the y-coordinate reveals that it is not the x-intercept. This option serves as a valuable lesson in the importance of thoroughness and attention to detail when solving mathematical problems. It's not enough to find one part of the answer; one must ensure that all conditions are met for the answer to be correct. This principle applies not only to x-intercept problems but to a wide range of mathematical and scientific problems.

Conclusion

In conclusion, finding the x-intercept of the function g(x) = 3x - 6 involves setting g(x) to zero and solving for x, which yields x = 2. The x-intercept is the point (2, 0), corresponding to option D, (2, g(2)). Understanding why the other options are incorrect reinforces the concept of x-intercepts and helps prevent future errors. This exercise demonstrates a fundamental skill in algebra and its application in understanding the behavior of functions and their graphs. The ability to find x-intercepts is not only crucial for solving mathematical problems but also for interpreting real-world scenarios modeled by functions. Whether it's determining break-even points in economics or finding the roots of a polynomial in calculus, the concept of x-intercepts plays a vital role. Furthermore, the process of analyzing incorrect options is equally important as it deepens the understanding of the underlying principles and helps in avoiding common pitfalls. By understanding why certain answers are incorrect, students can develop a more robust and nuanced understanding of the topic. This approach not only improves problem-solving skills but also fosters a deeper appreciation for the mathematical concepts involved. Therefore, mastering the skill of finding x-intercepts, along with the ability to critically evaluate different options, is essential for success in mathematics and its applications in various fields.