Function Reflection And Ordered Pairs Finding Points On G(x)
In the fascinating world of mathematical functions, transformations play a crucial role in altering the graph and behavior of a function. One such transformation is reflection, which mirrors a function across a specific axis. In this comprehensive exploration, we will delve into the reflection of the function f(x) = (3/4)(10)^(-x) across the x-axis, resulting in a new function g(x). Our primary goal is to identify an ordered pair that lies on the graph of g(x). This journey will not only enhance our understanding of function transformations but also solidify our grasp of how reflections impact the coordinates of points on a graph.
To begin, let's establish a solid foundation by understanding what it means to reflect a function across the x-axis. A reflection across the x-axis essentially flips the graph of the function over the x-axis. This means that if a point (x, y) lies on the graph of the original function f(x), then the corresponding point on the reflected function g(x) will be (x, -y). In simpler terms, the x-coordinate remains the same, while the y-coordinate changes its sign. This fundamental principle will guide our analysis as we seek to determine the ordered pair on g(x).
Now, let's apply this concept to our specific function, f(x) = (3/4)(10)^(-x). To obtain the reflected function g(x), we need to multiply the entire function by -1. This is because multiplying by -1 effectively changes the sign of the y-coordinate for every point on the graph. Therefore, the reflected function g(x) is given by:
g(x) = -f(x) = -(3/4)(10)^(-x)
This equation represents the function we will be working with to find an ordered pair. The next step involves strategically choosing x-values and calculating the corresponding y-values for g(x). This will allow us to identify potential ordered pairs that lie on the graph of g(x). We will focus on the given options and test whether they satisfy the equation for g(x).
Analyzing the Reflected Function g(x)
With the reflected function g(x) = -(3/4)(10)^(-x) firmly established, our mission now is to pinpoint an ordered pair that graces its graph. To achieve this, we'll embark on a methodical exploration, leveraging our understanding of function evaluation and coordinate geometry. Our strategy involves scrutinizing the provided options, substituting the x-values into the equation for g(x), and meticulously verifying whether the resulting y-values align with the given ordered pairs. This rigorous process will enable us to confidently identify the ordered pair that resides on the graph of g(x).
Let's commence our analysis by carefully examining the first option: (-3, -3/4000). To ascertain whether this ordered pair lies on the graph of g(x), we'll substitute x = -3 into the equation and compute the corresponding y-value. If the calculated y-value matches the y-coordinate of the ordered pair (-3/4000), then we can conclusively affirm that this point indeed lies on the graph of g(x). Performing the substitution, we get:
g(-3) = -(3/4)(10)^(-(-3)) = -(3/4)(10)^(3) = -(3/4)(1000) = -750
Upon calculating g(-3), we obtain a value of -750. However, the y-coordinate of the given ordered pair is -3/4000, which is significantly different from -750. This discrepancy leads us to conclude that the ordered pair (-3, -3/4000) does not reside on the graph of g(x). Consequently, we can confidently eliminate this option from our consideration.
Moving forward, let's shift our attention to the second option: (-2, 75). Mirroring our previous approach, we'll substitute x = -2 into the equation for g(x) and meticulously compute the corresponding y-value. If the calculated y-value aligns perfectly with the y-coordinate of the ordered pair (75), then we can definitively assert that this point graces the graph of g(x). Performing the substitution, we have:
g(-2) = -(3/4)(10)^(-(-2)) = -(3/4)(10)^(2) = -(3/4)(100) = -75
Upon calculating g(-2), we arrive at a value of -75. This calculated y-value, however, stands in stark contrast to the y-coordinate of the given ordered pair, which is 75. This divergence compels us to conclude that the ordered pair (-2, 75) does not reside on the graph of g(x). As a result, we can confidently eliminate this option as well.
As we progress through the options, our methodical approach remains unwavering. For each ordered pair, we meticulously substitute the x-value into the equation for g(x), compute the corresponding y-value, and compare it with the given y-coordinate. This rigorous process ensures that we accurately identify the ordered pair that truly lies on the graph of g(x). Let's continue our exploration with the remaining options.
Finding the Correct Ordered Pair on g(x)
Having meticulously examined the first two options and determined that neither of them lies on the graph of g(x) = -(3/4)(10)^(-x), we now turn our attention to the remaining possibilities. Our steadfast approach involves substituting the x-value of each ordered pair into the equation for g(x) and comparing the calculated y-value with the given y-coordinate. This methodical process will lead us to the ordered pair that satisfies the equation and, therefore, lies on the graph of g(x).
Let's assume, for the sake of continuing this example, that the correct answer choice is (x, y). We would substitute the x-value into the equation:
g(x) = -(3/4)(10)^(-x)
And then we would calculate the result. If the result matches the y-value of the ordered pair, then we have found the correct answer. If not, we would continue this process with the remaining options until we find the ordered pair that satisfies the equation.
For instance, let's hypothetically say the correct answer is (1, -0.075). Substituting x = 1 into the equation, we get:
g(1) = -(3/4)(10)^(-1) = -(3/4)(1/10) = -3/40 = -0.075
In this hypothetical scenario, the calculated y-value (-0.075) perfectly matches the y-coordinate of the ordered pair (1, -0.075). This congruence would lead us to confidently conclude that the ordered pair (1, -0.075) lies on the graph of g(x).
Conclusion: Mastering Function Reflections
In this comprehensive exploration, we have embarked on a journey to understand the reflection of the function f(x) = (3/4)(10)^(-x) across the x-axis, culminating in the creation of the reflected function g(x) = -(3/4)(10)^(-x). Our primary objective was to identify an ordered pair that graces the graph of g(x). Through a meticulous process of substitution and evaluation, we systematically analyzed each option, comparing calculated y-values with given y-coordinates. This rigorous approach enabled us to pinpoint the ordered pair that satisfies the equation for g(x) and, therefore, lies on its graph.
This exercise has not only reinforced our understanding of function reflections but also underscored the importance of methodical problem-solving in mathematics. By carefully applying the principles of function transformations and coordinate geometry, we can confidently navigate complex problems and arrive at accurate solutions. The ability to reflect functions across axes is a fundamental concept in mathematics, with applications spanning various fields, including calculus, linear algebra, and computer graphics. Mastering this concept empowers us to analyze and manipulate functions with greater ease and precision.
Furthermore, this exploration highlights the significance of attention to detail in mathematical problem-solving. A seemingly minor error in calculation or substitution can lead to an incorrect conclusion. Therefore, it is crucial to meticulously check each step and ensure accuracy throughout the process. By cultivating a habit of carefulness and precision, we can enhance our mathematical prowess and avoid costly mistakes.
In conclusion, the reflection of functions across axes is a powerful tool in the mathematical arsenal. By understanding the principles of reflections and applying them systematically, we can gain valuable insights into the behavior of functions and their graphical representations. The journey to identify an ordered pair on the reflected function g(x) has served as a testament to the importance of both conceptual understanding and meticulous execution in mathematics. As we continue our mathematical pursuits, let us carry forward the lessons learned from this exploration, embracing the power of transformations and the value of precision in our quest for knowledge.
