Function Reflection Find Ordered Pair On G(x)

JU07/11/2025 08, 2025 by THE IDEN 46 views

Exploring Reflections and Exponential Functions

In the realm of mathematics, functions serve as fundamental tools for modeling relationships between variables. Exponential functions, in particular, play a crucial role in describing phenomena that exhibit rapid growth or decay. This article delves into the intricacies of exponential functions, focusing on the function f(x) = (1/6)(2/5)^x and its reflection across the y-axis. We will explore the properties of this function, understand how reflections transform it, and determine specific points that lie on the reflected function.

Understanding Exponential Functions

At its core, an exponential function is defined by the general form f(x) = a * b^x, where a is the initial value and b is the base. The base b determines the function's behavior: if b is greater than 1, the function exhibits exponential growth; if b is between 0 and 1, the function exhibits exponential decay. In our case, f(x) = (1/6)(2/5)^x has an initial value of 1/6 and a base of 2/5. Since 2/5 is between 0 and 1, this function represents exponential decay. As x increases, the value of f(x) decreases, approaching zero but never actually reaching it. This behavior is characteristic of exponential decay functions. The constant factor of 1/6 in our function serves to scale the exponential term. It vertically compresses the function, making the decay less pronounced compared to the function (2/5)^x. This scaling factor plays a crucial role in determining the specific values of the function for different inputs.

To fully grasp the behavior of f(x), let's examine its key characteristics. The domain of f(x) is all real numbers, meaning that we can input any real number for x. The range of f(x) is (0, ∞), indicating that the function's output is always positive and never reaches zero. This is a hallmark of exponential functions – they never cross the x-axis. The y-intercept of f(x) is the point where the function intersects the y-axis, which occurs when x = 0. Substituting x = 0 into the function, we get f(0) = (1/6)(2/5)^0 = 1/6. Therefore, the y-intercept is (0, 1/6). As x approaches positive infinity, f(x) approaches zero, indicating the exponential decay. As x approaches negative infinity, f(x) approaches infinity, showing the unbounded growth in the opposite direction. The graph of f(x) is a smooth curve that starts high on the left side, gradually decreases as x increases, and approaches the x-axis as it extends to the right. Understanding these characteristics provides a solid foundation for analyzing transformations of the function.

Reflections Across the Y-Axis

Reflecting a function across the y-axis is a fundamental transformation that alters its symmetry. Geometrically, a reflection across the y-axis can be visualized as flipping the graph of the function horizontally over the y-axis. This transformation effectively reverses the x-coordinates of all points on the graph while preserving the y-coordinates. Mathematically, reflecting a function f(x) across the y-axis results in a new function g(x), where g(x) = f(-x). This means that to find the reflected function, we simply replace every x in the original function with -x. This substitution has a profound impact on the function's behavior, particularly for exponential functions.

In our specific case, reflecting f(x) = (1/6)(2/5)^x across the y-axis results in the function g(x) = f(-x) = (1/6)(2/5)^(-x). To simplify this expression, we can use the property that a^(-x) = (1/a)^x. Applying this property to our function, we get g(x) = (1/6)(5/2)^x. Notice that the base of the exponential term has changed from 2/5 to 5/2. This seemingly small change has a significant impact on the function's behavior. Since 5/2 is greater than 1, g(x) represents exponential growth, whereas f(x) represented exponential decay. The reflection across the y-axis has effectively transformed a decaying exponential function into a growing one. This transformation highlights the sensitivity of exponential functions to changes in the base and the power of reflections in altering function behavior. Understanding this transformation is crucial for solving problems involving reflections and exponential functions.

Determining Points on g(x)

Now that we have the reflected function, g(x) = (1/6)(5/2)^x, we can determine which ordered pairs lie on its graph. To do this, we substitute the x-coordinate of each ordered pair into the function g(x) and check if the resulting y-coordinate matches the ordered pair's y-coordinate. This process involves evaluating the exponential expression (5/2)^x for different values of x and then scaling the result by the factor 1/6. Let's examine the given options:

A. (-3, 4/375): Substituting x = -3 into g(x), we get g(-3) = (1/6)(5/2)^(-3) = (1/6)(2/5)^3 = (1/6)(8/125) = 4/375. Therefore, the ordered pair (-3, 4/375) lies on g(x). This ordered pair satisfies the equation of the reflected function, indicating that it is a valid point on the graph.

B. (-2, 25/24): Substituting x = -2 into g(x), we get g(-2) = (1/6)(5/2)^(-2) = (1/6)(2/5)^2 = (1/6)(4/25) = 2/75. This does not match the y-coordinate of 25/24, so the ordered pair (-2, 25/24) does not lie on g(x). The discrepancy between the calculated y-value and the given y-value indicates that this point is not on the graph of the reflected function.

C. (2, 5/12): Substituting x = 2 into g(x), we get g(2) = (1/6)(5/2)^2 = (1/6)(25/4) = 25/24. This does not match the y-coordinate in option C.

Conclusion

In conclusion, by reflecting the exponential function f(x) = (1/6)(2/5)^x across the y-axis, we obtain the function g(x) = (1/6)(5/2)^x. By substituting the x-coordinates of the provided ordered pairs into g(x), we find that only the ordered pair (-3, 4/375) lies on the reflected function g(x). This exercise demonstrates the interplay between exponential functions, reflections, and function evaluation, providing a valuable insight into the world of mathematical transformations. Understanding these concepts is essential for tackling more complex mathematical problems and appreciating the elegance of function transformations.

Let's explore how reflecting a function across the y-axis transforms its equation and how to identify points on the transformed function. In this article, we'll focus on the specific function f(x) = (1/6)(2/5)^x and its reflection across the y-axis, denoted as g(x). Our goal is to determine which of the given ordered pairs lies on the graph of g(x).

Understanding the Reflection Transformation

Reflecting a function across the y-axis is a geometric transformation that creates a mirror image of the original function with respect to the y-axis. This transformation has a specific mathematical effect on the function's equation. If we have a function f(x), its reflection across the y-axis, g(x), is given by g(x) = f(-x). This means that to obtain the equation for g(x), we replace every instance of x in the equation for f(x) with -x. This simple substitution is the key to understanding how reflections affect functions.

In our case, we start with the exponential function f(x) = (1/6)(2/5)^x. To find its reflection g(x), we replace x with -x: g(x) = f(-x) = (1/6)(2/5)^(-x). Now, let's simplify this expression. Recall that a^(-x) is equivalent to (1/a)^x. Applying this rule, we get g(x) = (1/6)((2/5)^(-1))x = (1/6)(5/2)^x. Thus, the reflected function is g(x) = (1/6)(5/2)^x. Notice how the base of the exponential term has changed from 2/5 in f(x) to 5/2 in g(x). This change is crucial, as it transforms the function from representing exponential decay to exponential growth. f(x) decreases as x increases, while g(x) increases as x increases. This difference in behavior is a direct consequence of the reflection across the y-axis.

The reflection across the y-axis fundamentally changes the nature of the exponential function. The original function, f(x), is an exponential decay function because the base (2/5) is between 0 and 1. As x increases, the function values get smaller and smaller, approaching zero. The reflected function, g(x), is an exponential growth function because the base (5/2) is greater than 1. As x increases, the function values get larger and larger. This transformation highlights the symmetry created by the reflection and the impact of changing the sign of the exponent.

Identifying Points on g(x)

To determine which of the given ordered pairs lies on g(x), we substitute the x-coordinate of each ordered pair into the equation for g(x) and see if the resulting y-coordinate matches the y-coordinate of the ordered pair. This is the fundamental way to check if a point lies on a function's graph – the point's coordinates must satisfy the function's equation. Let's go through each option:

Option A: (-3, 4/375)

Substitute x = -3 into g(x) = (1/6)(5/2)^x: g(-3) = (1/6)(5/2)^(-3). To evaluate this, we rewrite (5/2)^(-3) as (2/5)^3, which is (2^3)/(53) = 8/125. So, g(-3) = (1/6)(8/125) = 8/750. Simplifying the fraction, we get g(-3) = 4/375. This matches the y-coordinate of the ordered pair (-3, 4/375). Therefore, the ordered pair (-3, 4/375) lies on g(x). This result confirms that this point is indeed a solution to the equation g(x) = (1/6)(5/2)^x. We can confidently conclude that this point is on the graph of the reflected function.
Option B: (-2, 25/24)

Substitute x = -2 into g(x) = (1/6)(5/2)^x: g(-2) = (1/6)(5/2)^(-2). Rewriting (5/2)^(-2) as (2/5)^2, we get (2^2)/(52) = 4/25. So, g(-2) = (1/6)(4/25) = 4/150. Simplifying the fraction, we get g(-2) = 2/75. This does not match the y-coordinate of 25/24. Therefore, the ordered pair (-2, 25/24) does not lie on g(x). This discrepancy indicates that this point is not a solution to the function's equation and therefore is not on the graph.
Option C: (2, 5/12)

Substitute x = 2 into g(x) = (1/6)(5/2)^x: g(2) = (1/6)(5/2)^2. Evaluating (5/2)^2, we get (5^2)/(22) = 25/4. So, g(2) = (1/6)(25/4) = 25/24. This does not match the y-coordinate of 5/12. Therefore, the ordered pair (2, 5/12) does not lie on g(x). This mismatch confirms that this point is not a valid solution for the function and does not lie on the graph.

Conclusion

After analyzing each option, we've determined that the ordered pair (-3, 4/375) is the only one that lies on the graph of the reflected function g(x) = (1/6)(5/2)^x. This exercise demonstrates the importance of understanding function transformations, particularly reflections, and how to apply them to find points on the transformed function. By carefully substituting x-coordinates into the function's equation and comparing the results with the given y-coordinates, we can accurately determine whether a point lies on the graph of a function. This skill is essential for solving a wide range of mathematical problems involving functions and their transformations.

In mathematics, understanding how transformations affect functions is a crucial skill. Among these transformations, reflections hold a special place due to their ability to create mirror images of functions across specific axes. In this comprehensive exploration, we will delve into the concept of reflecting a function across the y-axis and how to determine ordered pairs that lie on the reflected function. Our focus will be on the function f(x) = (1/6)(2/5)^x and its reflection across the y-axis, which we will denote as g(x). We will systematically identify the ordered pair that belongs to g(x), providing a step-by-step guide for similar problems.

The Essence of Function Reflection Across the Y-Axis

Function reflection across the y-axis is a geometric transformation that mirrors a function's graph over the y-axis. This transformation has a profound impact on the function's equation and behavior. When a function f(x) is reflected across the y-axis, the resulting function, g(x), is defined as g(x) = f(-x). This concise mathematical expression encapsulates the core concept of reflection: we replace every instance of x in the original function with its negation, -x. This substitution is the key to deriving the equation of the reflected function.

Let's apply this principle to our function, f(x) = (1/6)(2/5)^x. To find its reflection, g(x), we substitute -x for x: g(x) = f(-x) = (1/6)(2/5)^(-x). Now, we can simplify this expression using the property of exponents that states a^(-x) = (1/a)^x. Applying this rule, we get g(x) = (1/6)((2/5)^(-1))x = (1/6)(5/2)^x. Thus, the reflected function is g(x) = (1/6)(5/2)^x. Notice the transformation in the base of the exponential term. In f(x), the base is 2/5, which is less than 1, indicating exponential decay. In g(x), the base is 5/2, which is greater than 1, indicating exponential growth. This change in the base is a direct consequence of the reflection and fundamentally alters the function's behavior.

To further illustrate the impact of reflection, consider the graphical representation of f(x) and g(x). The graph of f(x) is a decreasing exponential curve that approaches the x-axis as x increases. The graph of g(x), on the other hand, is an increasing exponential curve that rises rapidly as x increases. The y-axis acts as a mirror, with the two graphs being symmetrical about it. This symmetry highlights the visual aspect of reflection and reinforces the mathematical transformation we applied. Understanding this connection between the graphical and algebraic representations is crucial for a comprehensive understanding of function reflections.

Determining Ordered Pairs on the Reflected Function

Now that we have derived the equation for the reflected function, g(x) = (1/6)(5/2)^x, we can proceed to identify ordered pairs that lie on its graph. An ordered pair (x, y) lies on the graph of a function if and only if the coordinates satisfy the function's equation, meaning that g(x) = y. To check if an ordered pair belongs to g(x), we substitute the x-coordinate into the equation for g(x) and verify if the resulting value matches the y-coordinate of the ordered pair. This is a fundamental technique in coordinate geometry and a powerful tool for analyzing functions.

Let's apply this method to the provided options:

Option A: (-3, 4/375)

Substitute x = -3 into g(x) = (1/6)(5/2)^x: g(-3) = (1/6)(5/2)^(-3). Recall that a^(-n) = 1/(a^n). Thus, (5/2)^(-3) = 1/((5/2)^3) = 1/(125/8) = 8/125. Therefore, g(-3) = (1/6)(8/125) = 8/750. Simplifying the fraction, we get g(-3) = 4/375. This value matches the y-coordinate of the ordered pair (-3, 4/375). Hence, the ordered pair (-3, 4/375) lies on the graph of g(x). This confirmation establishes that this point is a solution to the equation g(x) = (1/6)(5/2)^x.
Option B: (-2, 25/24)

Substitute x = -2 into g(x) = (1/6)(5/2)^x: g(-2) = (1/6)(5/2)^(-2). Using the property a^(-n) = 1/(a^n), we have (5/2)^(-2) = 1/((5/2)^2) = 1/(25/4) = 4/25. Therefore, g(-2) = (1/6)(4/25) = 4/150. Simplifying the fraction, we get g(-2) = 2/75. This value does not match the y-coordinate of 25/24. Therefore, the ordered pair (-2, 25/24) does not lie on the graph of g(x). The mismatch between the calculated value and the given y-coordinate indicates that this point is not a solution to the equation.
Option C: (2, 5/12)

Substitute x = 2 into g(x) = (1/6)(5/2)^x: g(2) = (1/6)(5/2)^2. Evaluating (5/2)^2, we get 25/4. Therefore, g(2) = (1/6)(25/4) = 25/24. This value does not match the y-coordinate of 5/12. Therefore, the ordered pair (2, 5/12) does not lie on the graph of g(x). The difference between the calculated value and the given y-coordinate confirms that this point is not on the function's graph.

Conclusion: The Ordered Pair on the Reflected Function

Through our detailed analysis, we have conclusively determined that the ordered pair (-3, 4/375) is the only one that lies on the graph of the reflected function g(x) = (1/6)(5/2)^x. This comprehensive exploration has highlighted the significance of understanding function reflections, their impact on equations, and the method for identifying ordered pairs on transformed functions. By mastering these concepts, one gains a deeper appreciation for the elegance and power of mathematical transformations. The ability to analyze function transformations is not only crucial for solving specific problems but also for developing a broader understanding of mathematical relationships and their applications in various fields.