Finding Points On Reflected Functions A Step-by-Step Guide

Understanding functions is the cornerstone of mathematical analysis, and transformations of these functions offer profound insights into their behavior. One common transformation is reflection across the x-axis. When a function is reflected across the x-axis, its graph is flipped over the x-axis, creating a mirror image. This transformation affects the y-coordinates of the points on the graph while leaving the x-coordinates unchanged. In mathematical terms, if a point (x, y) lies on the graph of the original function, then the point (x, -y) will lie on the graph of the reflected function. Understanding this concept is crucial for analyzing the properties and behavior of various functions, especially in fields like physics, engineering, and computer graphics, where transformations play a vital role in modeling real-world phenomena. This transformation changes the sign of the function's output, effectively inverting the graph vertically. Let's delve into the specifics of reflecting the function f(x) = (3/4)(10)^(-x) across the x-axis. This type of transformation is fundamental in understanding how functions behave under different operations and is a key concept in precalculus and calculus. In the world of mathematical transformations, reflections stand as a cornerstone concept, especially when dissecting functions. To reflect a function across the x-axis, we're essentially creating a mirror image of the function over this axis. This transformation profoundly impacts the function's graphical representation and algebraic expression, which we'll explore in detail. Reflections are not just abstract mathematical operations; they're vital in various scientific and technological fields. For instance, in physics, understanding how light reflects is crucial in optics. Similarly, in computer graphics, reflections are used to create realistic images and animations. Therefore, mastering the concept of reflections is essential for anyone venturing into STEM fields. The reflection of a function is a transformation that flips the graph of the function over a line, which in this case is the x-axis.

Defining the Original Function: f(x) = (3/4)(10)^(-x)

Before we explore the reflection, it's important to dissect the original function, f(x) = (3/4)(10)^(-x). This is an exponential function with a base of 10 and a negative exponent, indicating that the function decreases as x increases. The coefficient (3/4) scales the function vertically. To truly grasp the behavior of this function, we need to consider its key characteristics. Exponential functions like this one play a crucial role in modeling various phenomena in the real world, from population growth to radioactive decay. The negative exponent signifies that we are dealing with exponential decay, where the function's value diminishes as x grows. The (3/4) factor acts as a vertical compression, scaling the function's output values. These functions are characterized by a rapid change in their values, making them indispensable tools in fields such as finance, biology, and physics. For instance, in finance, exponential decay models the depreciation of assets, while in biology, it can represent the elimination of a drug from the body. This function is defined for all real numbers, meaning that we can input any value for x. However, its output is always positive, approaching zero as x approaches infinity and increasing rapidly as x becomes more negative. This behavior is characteristic of exponential decay functions, where the rate of decrease is proportional to the current value. Understanding these properties will help us predict how the function will change when reflected across the x-axis. In this particular function, the base of the exponent is 10, which influences the rate of decay. A larger base leads to a faster decay rate. The coefficient 3/4, on the other hand, affects the initial value of the function at x = 0. This constant acts as a vertical compression, scaling the function's output values by a factor of 3/4. In mathematical modeling, these coefficients often represent initial conditions or scaling factors that adapt the function to real-world data. Understanding how each component contributes to the overall behavior of the function is crucial for applying this function to solve real-world problems and for predicting its transformation under various operations. For instance, knowing that the function decreases as x increases, we can anticipate that its reflection will increase as x increases.

Reflecting Across the x-axis: Deriving g(x)

To reflect f(x) across the x-axis, we negate the entire function. This means that g(x) = -f(x). Substituting f(x), we get g(x) = -(3/4)(10)^(-x). This transformation flips the graph of f(x) over the x-axis. When a function is reflected across the x-axis, all the y-coordinates change their sign. Positive y-values become negative, and negative y-values become positive, while the x-coordinates remain unchanged. This process essentially creates a mirror image of the function with respect to the x-axis. Understanding this transformation is crucial in various mathematical contexts, including analyzing the symmetry of functions and solving equations involving reflections. In geometric terms, a reflection is a transformation that maps each point of a figure to its mirror image with respect to a given line, which in this case is the x-axis. In the context of functions, this transformation can be represented algebraically by negating the function's output. This concept is not only fundamental in mathematics but also has applications in fields such as physics and computer graphics, where reflections are used to model real-world phenomena and create visual effects. Once we have derived the equation for g(x), we can analyze its properties and behavior in comparison to f(x). The reflection across the x-axis changes the sign of the output, but it does not affect the domain of the function. This means that g(x) is also defined for all real numbers. However, the range of g(x) is now the negative of the range of f(x). Since f(x) is always positive, g(x) is always negative. Furthermore, the behavior of g(x) is the opposite of that of f(x). While f(x) decreases as x increases, g(x) increases as x increases. This difference in behavior is a direct consequence of the reflection across the x-axis and is a key characteristic of this type of transformation. The function g(x) is obtained by simply negating the function f(x). This means that if (x, y) is a point on the graph of f(x), then (x, -y) is a point on the graph of g(x).

Identifying a Point on g(x)

Now, let's examine the provided ordered pairs to determine which one lies on g(x) = -(3/4)(10)^(-x). To do this, we'll substitute the x-coordinate of each pair into g(x) and check if the result matches the y-coordinate. This process of verifying points on a function's graph is a fundamental skill in algebra and calculus. It involves substituting the x-coordinate into the function's equation and comparing the calculated y-value with the given y-coordinate. If the two values match, the point lies on the graph; otherwise, it does not. This technique is not only useful for verifying points but also for plotting graphs and understanding the behavior of functions. In many real-world applications, we often need to determine whether a particular data point fits a given mathematical model. Verifying points on a graph is a crucial step in this process, allowing us to validate the model and make predictions. In the context of the reflection we've discussed, we're essentially checking if the reflected function aligns with the transformed points. This process reinforces the understanding of how transformations affect the coordinates of points on a graph. Verifying points on a function's graph is a fundamental skill in mathematics, bridging the gap between abstract equations and their visual representations. When given an ordered pair and a function, the process is straightforward: substitute the x-coordinate from the ordered pair into the function and see if the resulting y-value matches the y-coordinate in the ordered pair. This technique is not just a mechanical exercise; it deeply connects the algebraic definition of a function with its geometric representation. Each point on a function's graph represents a solution to the function's equation, and this verification process confirms whether a particular point satisfies the equation. Moreover, this skill is invaluable in practical applications, such as validating the fit of a mathematical model to empirical data or verifying the correctness of graphical transformations. The ability to accurately verify points on a graph is a cornerstone of mathematical proficiency, opening doors to deeper understanding and application of mathematical concepts in various fields. This is a hands-on way to connect the algebraic representation of a function with its graphical representation. Each point on the graph of a function is a solution to the function's equation. By substituting the x-coordinate of a point into the function, we can calculate the corresponding y-coordinate. If the calculated y-coordinate matches the given y-coordinate, then the point lies on the graph of the function. This process is not just about plugging in numbers; it's about understanding the relationship between the input (x) and the output (y) of the function. It's a fundamental skill that is used in various contexts, such as plotting graphs, solving equations, and analyzing data.

Testing Option A: (-3, -3/4000)

Substitute x = -3 into g(x): g(-3) = -(3/4)(10)^(-(-3)) = -(3/4)(10)^(3) = -(3/4)(1000) = -750. Since -750 does not equal -3/4000, this point is not on g(x). This is a critical step in mathematical problem-solving: systematically testing each option to arrive at the correct answer. In this case, we're applying the same principle we discussed earlier—substituting the x-coordinate into the function's equation and comparing the result with the y-coordinate. The careful execution of this step is crucial to avoid errors and ensure the accuracy of the solution. Testing each option involves a combination of algebraic manipulation and arithmetic calculation. It also requires a clear understanding of the function's properties and how it behaves under different inputs. For instance, in this case, we're dealing with an exponential function, and the negative exponent can sometimes lead to confusion if not handled correctly. Therefore, each step in the calculation must be performed with precision and attention to detail. This process is not just about finding the right answer; it's also about developing a methodical approach to problem-solving, which is a valuable skill in any field. The substitution process itself involves several steps, each of which must be executed carefully. First, we replace the variable x in the function's equation with the x-coordinate of the ordered pair. Then, we simplify the expression using the order of operations (PEMDAS/BODMAS). This often involves dealing with exponents, multiplication, and division. Finally, we compare the calculated y-value with the y-coordinate of the ordered pair. If the two values match, then the point lies on the graph of the function; otherwise, it does not. This process may seem straightforward, but it requires practice and attention to detail to avoid errors. The process of substituting values and evaluating the function is a cornerstone of mathematical problem-solving. It bridges the gap between abstract equations and concrete numerical results. The careful substitution of x = -3 into g(x) is not merely a mechanical task; it is an opportunity to reinforce the understanding of exponential functions and their properties. Each step of the calculation, from handling the negative exponent to performing the arithmetic operations, must be executed with precision.

Testing Option B: (-2, -75)

Substitute x = -2 into g(x): g(-2) = -(3/4)(10)^(-(-2)) = -(3/4)(10)^(2) = -(3/4)(100) = -75. Since -75 equals -75, this point is on g(x). This demonstrates the importance of thoroughness and accuracy in mathematical problem-solving. By carefully substituting the x-coordinate into the function and evaluating the expression, we can determine whether the point lies on the graph of the function. In this case, the calculated y-value matched the given y-coordinate, confirming that the point (-2, -75) is indeed on g(x). This process not only provides the correct answer but also reinforces the understanding of how functions work and how their graphs are represented. The substitution process is a fundamental tool in mathematics, and it is used in a wide variety of contexts. From solving equations to graphing functions, the ability to substitute values and evaluate expressions is essential. The process of substituting values into functions and evaluating the resulting expressions is a fundamental skill in mathematics. The accurate calculation of g(-2) confirms the point (-2, -75) lies on the graph of g(x). This exercise is a microcosm of the broader mathematical process, which involves applying definitions, performing calculations, and drawing conclusions. In this instance, the correct conclusion not only answers the specific question but also reinforces the understanding of function evaluation and graphing. This is a practical application of the concept of function evaluation.

Final Answer

The ordered pair (-2, -75) is on g(x).

Which ordered pair lies on the function g(x), which is the reflection of f(x) = (3/4)(10)^(-x) across the x-axis?

Finding Points on Reflected Functions A Step-by-Step Guide