Exponential Function F(x) = 4(5)^(x-3) Calculation And Applications

In the realm of mathematics, exponential functions hold a prominent place, modeling phenomena that exhibit rapid growth or decay. These functions are characterized by a constant base raised to a variable exponent, creating curves that either soar upwards or plummet downwards. This article delves into the intricacies of a specific exponential function, f(x) = 4(5)^(x-3), exploring its properties, behavior, and practical applications. Our primary focus will be on calculating f(5), providing a step-by-step guide to understanding how to evaluate exponential functions for a given input. Understanding exponential functions is crucial in various fields, from finance and biology to physics and computer science. They help us model population growth, radioactive decay, compound interest, and many other real-world phenomena. The function f(x) = 4(5)^(x-3), the subject of this exploration, exemplifies the core characteristics of exponential functions. By manipulating the base, exponent, and coefficient, we can tailor these functions to represent a wide array of scenarios. This detailed analysis will not only provide a solution for f(5) but also equip you with a comprehensive understanding of exponential functions, empowering you to tackle similar problems with confidence. In the following sections, we will dissect the components of the function, analyze its graphical representation, and discuss its implications in practical contexts. So, let's embark on this mathematical journey, unraveling the mysteries of exponential functions and gaining mastery over their application.

Understanding the Exponential Function: f(x) = 4(5)^(x-3)

At the heart of our exploration lies the exponential function f(x) = 4(5)^(x-3). To fully grasp its behavior, we must dissect its components and understand how each element contributes to the overall function. The number 5 serves as the base of the exponential term. The base dictates the rate at which the function grows or decays. In this case, since 5 is greater than 1, the function exhibits exponential growth. This means that as x increases, the value of f(x) increases rapidly. The term (x-3) in the exponent is another crucial element. This term represents a horizontal shift of the function. Subtracting 3 from x shifts the graph 3 units to the right. This shift affects the function's initial value and its overall position on the coordinate plane. The coefficient 4, which multiplies the exponential term, acts as a vertical stretch factor. It scales the function vertically, affecting its amplitude. In simpler terms, it determines how much the function is stretched or compressed along the y-axis. Together, these components – the base, the exponent, and the coefficient – orchestrate the unique behavior of the exponential function. By understanding each element's role, we can predict how the function will behave for different values of x. For instance, we know that as x becomes very large, the exponential term (5)^(x-3) will dominate, causing f(x) to grow rapidly. Conversely, as x becomes very small (large negative values), the exponential term will approach zero, and f(x) will approach zero as well. This detailed understanding forms the foundation for solving the core problem of this article: calculating f(5). By carefully substituting x = 5 into the function, we can leverage our knowledge of each component to arrive at the correct answer. This process not only provides a numerical solution but also reinforces our understanding of how exponential functions work.

Step-by-Step Calculation of f(5)

Now, let's embark on the step-by-step calculation of f(5) for the exponential function f(x) = 4(5)^(x-3). This process will demonstrate how to apply our understanding of the function's components to arrive at a precise numerical solution. The first and most crucial step is to substitute x = 5 into the function. This means replacing every instance of x in the equation with the number 5. This substitution transforms the function from a general expression f(x) to a specific value f(5). After the substitution, our equation becomes f(5) = 4(5)^(5-3). This equation now represents a concrete numerical expression that we can evaluate. The next step involves simplifying the exponent. We have (5-3) in the exponent, which simplifies to 2. This simplification is essential because it allows us to work with a manageable power of the base. Our equation now looks like f(5) = 4(5)^2. We have reduced the exponential term to a simple power of 5. Now, we need to evaluate the exponential term. 5^2 means 5 multiplied by itself, which equals 25. This step is a fundamental arithmetic operation that transforms the exponential term into a numerical value. Our equation is now f(5) = 4(25). We are nearing the final step of the calculation. The final step is to perform the multiplication. We have 4 multiplied by 25, which equals 100. This final arithmetic operation yields the value of f(5). Therefore, we conclude that f(5) = 100. This result represents the value of the exponential function f(x) = 4(5)^(x-3) when x is equal to 5. By following these step-by-step calculations, we have not only arrived at the solution but also reinforced the process of evaluating exponential functions for specific inputs. This method can be applied to any exponential function, providing a clear and systematic approach to finding the function's value at a given point. The result, f(5) = 100, gives us a specific point on the graph of the exponential function, which can be further used to analyze the behavior and characteristics of the function.

Visualizing the Function: Graph and Key Features

To gain a deeper understanding of the exponential function f(x) = 4(5)^(x-3), visualizing its graph is invaluable. The graph provides a visual representation of the function's behavior, allowing us to identify key features such as its growth rate, intercepts, and asymptotes. Exponential functions, in general, exhibit a characteristic curve that either rises rapidly (exponential growth) or falls rapidly (exponential decay). In the case of f(x) = 4(5)^(x-3), since the base (5) is greater than 1, the function demonstrates exponential growth. This means that as x increases, the value of f(x) increases at an accelerating rate. The graph starts slowly but then climbs steeply upwards. One of the crucial features to identify is the y-intercept. The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function: f(0) = 4(5)^(0-3) = 4(5)^(-3) = 4(1/125) = 4/125. Therefore, the y-intercept is at the point (0, 4/125). This point provides a starting reference for the function's value. Another important feature is the asymptote. An asymptote is a line that the graph approaches but never quite touches. For exponential functions of the form f(x) = a(b)^(x-c) + d, the horizontal asymptote is typically the line y = d. In our case, f(x) = 4(5)^(x-3) can be written as f(x) = 4(5)^(x-3) + 0, so the horizontal asymptote is the line y = 0 (the x-axis). This means that as x becomes increasingly negative, the graph approaches the x-axis but never crosses it. The horizontal shift caused by the (x-3) term in the exponent also affects the graph's position. It shifts the entire graph 3 units to the right. This shift means that the graph's characteristic exponential growth starts further along the x-axis compared to a basic exponential function like f(x) = 5^x. By visualizing the graph of f(x) = 4(5)^(x-3), we gain a holistic understanding of its behavior. We can see how the exponential growth, y-intercept, and horizontal asymptote contribute to the overall shape and position of the curve. This visual representation complements our numerical calculations and helps us appreciate the function's properties more intuitively.

Applications of Exponential Functions in Real-World Scenarios

Exponential functions are not confined to the realm of pure mathematics; they play a crucial role in modeling various real-world phenomena. Their ability to represent rapid growth or decay makes them indispensable tools in diverse fields, from finance and biology to physics and computer science. One prominent application of exponential functions lies in the field of finance, specifically in modeling compound interest. Compound interest is the interest earned on both the principal amount and the accumulated interest. The formula for compound interest is given by A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. This formula is a classic example of an exponential function, where the base (1 + r/n) is raised to the power of nt. As time (t) increases, the final amount (A) grows exponentially, highlighting the power of compounding. In biology, exponential functions are used to model population growth. Under ideal conditions, populations of bacteria, animals, or even humans can grow exponentially. The population growth can be modeled by the equation N(t) = N₀e^(kt), where N(t) is the population at time t, N₀ is the initial population, e is the base of the natural logarithm (approximately 2.71828), and k is the growth rate constant. This equation demonstrates how populations can increase rapidly over time, provided there are sufficient resources and minimal constraints. Exponential functions are also essential in physics, particularly in modeling radioactive decay. Radioactive substances decay exponentially, meaning that the amount of substance decreases by a fixed percentage over a fixed period. The decay is modeled by the equation N(t) = N₀e^(-λt), where N(t) is the amount of substance remaining at time t, N₀ is the initial amount, e is the base of the natural logarithm, and λ is the decay constant. This equation helps scientists predict how much of a radioactive substance will remain after a certain time. In computer science, exponential functions appear in the analysis of algorithms. The time complexity of certain algorithms, particularly those involving searching or sorting, can be exponential. This means that the time required to run the algorithm increases exponentially with the size of the input. Understanding exponential functions is crucial for designing efficient algorithms and optimizing computational processes. These are just a few examples of the widespread applications of exponential functions. Their versatility and ability to model rapid change make them indispensable tools in many scientific and technological disciplines. By mastering the concepts and techniques associated with exponential functions, we can gain valuable insights into the world around us and solve complex problems in various domains.

Conclusion

In this comprehensive exploration, we have delved into the intricacies of the exponential function f(x) = 4(5)^(x-3). We began by dissecting the function's components, understanding the role of the base, exponent, and coefficient in shaping its behavior. This foundation allowed us to confidently calculate f(5), demonstrating a step-by-step process that can be applied to any exponential function. Our calculation revealed that f(5) = 100, a specific point on the function's graph. To further enhance our understanding, we visualized the function's graph, identifying key features such as its exponential growth, y-intercept, and horizontal asymptote. The graph provided a visual representation of the function's behavior, complementing our numerical calculations and offering a more intuitive grasp of its properties. Finally, we explored the wide-ranging applications of exponential functions in real-world scenarios. From modeling compound interest in finance to representing population growth in biology, radioactive decay in physics, and algorithm complexity in computer science, exponential functions prove to be indispensable tools across diverse disciplines. This exploration underscores the importance of mastering exponential functions, not only for mathematical proficiency but also for understanding and solving real-world problems. By grasping the concepts and techniques associated with these functions, we equip ourselves to analyze and model phenomena that exhibit rapid growth or decay, making informed decisions and contributing to advancements in various fields. The journey through f(x) = 4(5)^(x-3) has provided a valuable case study, illustrating the power and versatility of exponential functions. As we continue to encounter these functions in various contexts, the knowledge and skills gained here will serve as a solid foundation for further exploration and discovery. Mastering exponential functions opens doors to a deeper understanding of the world around us, empowering us to analyze, predict, and solve complex problems across a multitude of disciplines. This article serves as a stepping stone, encouraging further exploration and application of these powerful mathematical tools.