Finding F(7) For Exponential Function F(x) = Ab^x Given F(4.5) = 17 And F(5) = 94
Introduction to Exponential Functions
Exponential functions play a pivotal role in mathematics and various real-world applications. They elegantly model phenomena characterized by rapid growth or decay, such as population dynamics, compound interest, and radioactive decay. The general form of an exponential function is given by y = ab^x, where a represents the initial value, b is the base (growth or decay factor), and x is the independent variable. Understanding and manipulating exponential functions is crucial for solving problems across diverse scientific and engineering domains. When delving into the realm of mathematical functions, the exponential function stands out as a cornerstone concept with far-reaching applications. Defined by the form y = ab^x, where a and b are constants, and x is the variable exponent, this function beautifully encapsulates phenomena characterized by rapid growth or decay. Think of population dynamics, where the number of individuals can increase exponentially over time, or the decay of radioactive substances, where the quantity decreases exponentially. The constant a in the equation holds the key to the initial value, representing the function's output when x is zero. Meanwhile, the base b governs the rate of growth or decay. If b is greater than 1, the function exhibits exponential growth, soaring upwards as x increases. Conversely, if b is between 0 and 1, the function portrays exponential decay, gradually dwindling as x grows. Mastering the intricacies of exponential functions is not just an academic exercise; it's a gateway to understanding and modeling a vast array of real-world phenomena. From finance, where compound interest follows an exponential pattern, to biology, where population growth can be modeled exponentially, the applications are boundless. In physics, radioactive decay adheres to exponential decay, allowing scientists to date ancient artifacts. Engineering fields also heavily rely on exponential functions to analyze system stability and model signal propagation. The versatility and ubiquity of exponential functions make them indispensable tools for scientists, engineers, and mathematicians alike. As we embark on this mathematical journey, we'll not only dissect the equation y = ab^x but also explore the rich tapestry of applications woven around exponential functions. So, let's unlock the secrets of these functions and delve into their power to describe the world around us.
Problem Statement: Finding f(7)
In this particular problem, we are given an exponential function in the form f(x) = ab^x. We are provided with two data points: f(4.5) = 17 and f(5) = 94. Our mission is to determine the value of f(7) to the nearest tenth. This problem elegantly combines the theoretical understanding of exponential functions with practical problem-solving skills. The core challenge lies in using the given information to first determine the values of the constants a and b, which define the specific exponential function at hand. Once we have these constants, we can confidently evaluate f(7) and arrive at the solution. The beauty of this problem lies in its ability to test our comprehension of exponential functions and our ability to apply mathematical techniques to solve real-world problems. To precisely determine f(7) when dealing with an exponential function like f(x) = ab^x requires a methodical approach that harmoniously blends theoretical knowledge with practical problem-solving acumen. This particular problem presents us with a compelling challenge, as we're given two vital clues: f(4.5) = 17 and f(5) = 94. These seemingly simple data points hold the key to unlocking the mystery of our exponential function. The task at hand is to leverage this information to first decipher the values of a and b, the defining constants that shape the unique character of our function. Once these constants are revealed, we can confidently step forward and evaluate f(7), the ultimate goal of our mathematical quest. This problem isn't just about plugging numbers into an equation; it's about unveiling the intricate relationships within exponential functions. It demands a deeper understanding of how changes in x affect the function's output, guided by the interplay of the constants a and b. We'll need to carefully orchestrate our mathematical tools, such as substitution and algebraic manipulation, to extract the hidden values of a and b. Once we've accomplished this, evaluating f(7) will be a triumphant step, a testament to our mastery of exponential functions and problem-solving prowess. So, let's embark on this mathematical adventure with focus and determination, ready to dissect the problem, conquer the challenges, and emerge victorious with the value of f(7).
Step-by-Step Solution
1. Setting up the Equations
Using the given information, we can set up two equations:
- 17 = ab^(4.5)
- 94 = ab^5
These equations form a system of two equations with two unknowns (a and b). Solving this system will give us the values of a and b. The foundation of our solution lies in transforming the given information into a structured mathematical form. By expressing f(4.5) = 17 and f(5) = 94 as equations, we create a clear pathway to unraveling the constants a and b that define our exponential function. The act of setting up these equations isn't just a mechanical step; it's a critical translation of the problem's narrative into a language that our mathematical tools can understand. Each equation encapsulates a specific relationship between x and f(x), dictated by the underlying exponential function f(x) = ab^x. The constant a, representing the initial value, and the base b, governing the rate of growth, are the elusive pieces of the puzzle that we aim to uncover. With these two equations in hand, we've effectively laid the groundwork for a systematic solution. We're no longer dealing with abstract function values; we're now armed with concrete algebraic statements that can be manipulated and solved. The challenge now shifts to employing algebraic techniques to disentangle a and b from this system of equations. We might consider methods like substitution or elimination, carefully navigating the exponential nature of the equations. But regardless of the path we choose, the act of setting up these equations has transformed our problem into a tangible, solvable form, bringing us closer to the ultimate goal of finding f(7). Thus, let us proceed with the next steps, confident that our solid foundation will guide us to a successful resolution.
2. Solving for b
To eliminate a, we can divide the second equation by the first equation:
(94) / (17) = (ab^5) / (ab^(4.5))
This simplifies to:
94/17 = b^(5 - 4.5)
94/17 = b^(0.5)
To solve for b, we square both sides:
b = (94/17)^2
b ≈ 30.59
The elegance of this step lies in the strategic elimination of a, a move that simplifies our system of equations and allows us to focus solely on solving for b. By dividing the second equation by the first, we cleverly exploit the common factor of a, causing it to vanish from the equation. This maneuver not only reduces the complexity of the problem but also highlights the power of algebraic manipulation in unraveling mathematical relationships. The resulting equation, 94/17 = b^(0.5), is a significant milestone in our journey. It presents us with a direct relationship between the known quantity 94/17 and the unknown base b, raised to the power of 0.5. To isolate b, we employ a fundamental algebraic principle: squaring both sides of the equation. This operation effectively undoes the square root (0.5 power) and reveals the value of b. The calculated value, b ≈ 30.59, represents the growth factor of our exponential function. It tells us that for every unit increase in x, the function's value increases by a factor of approximately 30.59. This newfound knowledge is a crucial piece of the puzzle, bringing us closer to defining the complete exponential function. With b determined, we can now turn our attention to finding a, the initial value of the function. This will involve substituting the value of b back into one of our original equations and solving for a. The strategic elimination of a and the subsequent calculation of b exemplify the methodical approach required for solving problems involving exponential functions. Each step is carefully chosen to simplify the equations and reveal the hidden values that define the function's behavior. Thus, let's proceed with the next phase of our solution, building upon the foundation we've established and moving closer to the final answer.
3. Solving for a
Now that we have b, we can substitute it back into one of the original equations to solve for a. Let's use the first equation:
17 = a(30.59)^(4.5)
a = 17 / (30.59)^(4.5)
a ≈ 0.000088
The determination of b marked a significant step forward, and now we leverage that knowledge to unlock the value of a, the initial value of our exponential function. The strategic move here is to substitute the calculated value of b back into one of the original equations. This effectively transforms the equation into one with a single unknown, a, making it readily solvable. Choosing the first equation, 17 = a(30.59)^(4.5), is a matter of convenience, as either equation would ultimately lead to the same result. The subsequent steps involve isolating a through basic algebraic manipulation. We divide both sides of the equation by (30.59)^(4.5), effectively separating a from the known quantities. The resulting expression, a = 17 / (30.59)^(4.5), provides a direct pathway to calculating a. Evaluating this expression yields a remarkably small value for a, approximately 0.000088. This seemingly minuscule value holds significant meaning within the context of our exponential function. It represents the function's output when x is zero, the starting point from which the exponential growth begins. The small value of a, combined with the relatively large value of b (approximately 30.59), paints a vivid picture of a function that starts with a very small value but rapidly increases as x grows. With both a and b now determined, we have successfully unveiled the complete definition of our exponential function. We can confidently express it as f(x) = 0.000088 * (30.59)^x. This achievement marks a pivotal moment in our problem-solving journey, as we now possess the necessary tools to evaluate f(7), the ultimate goal we set out to achieve. Thus, let us proceed to the final step, where we'll put our newfound knowledge to work and calculate the value of f(7), rounding our answer to the nearest tenth.
4. Calculating f(7)
Now that we have a and b, we can find f(7):
f(7) = 0.000088 * (30.59)^7
f(7) ≈ 147148.4
The culmination of our efforts arrives at this final step, where we put our determined values of a and b to work and calculate f(7). With the exponential function fully defined as f(x) = 0.000088 * (30.59)^x, evaluating f(7) becomes a straightforward substitution. We replace x with 7 in the equation and perform the necessary calculations. The calculation involves raising 30.59 to the power of 7 and then multiplying the result by 0.000088. This computation reveals the value of f(7) to be approximately 147148.4. This significant value underscores the nature of exponential growth. Starting from a very small initial value (a ≈ 0.000088), the function rapidly increases as x grows, reaching a substantial value at x = 7. The magnitude of f(7) highlights the power of exponential functions to model phenomena characterized by rapid growth, where small changes in x can lead to dramatic changes in the function's output. The final instruction of the problem requires us to round our answer to the nearest tenth. Since the calculated value of f(7) is already given to the nearest tenth, no further rounding is necessary. Thus, we confidently present our final answer: f(7) ≈ 147148.4. This result represents the solution to our problem, the culmination of our step-by-step journey through the intricacies of exponential functions. We've successfully navigated the challenges, employed algebraic techniques, and ultimately determined the value of f(7). This achievement not only provides a numerical answer but also reinforces our understanding of exponential functions and their behavior.
Final Answer
Therefore, the value of f(7), to the nearest tenth, is approximately 147148.4.
Conclusion
This problem demonstrates the importance of understanding exponential functions and how to manipulate them. By setting up a system of equations and solving for the unknowns, we were able to find the value of the function at a specific point. The journey through this problem has not only provided us with a numerical answer but has also deepened our appreciation for the power and elegance of exponential functions. From setting up the initial equations to calculating the final value of f(7), each step has reinforced our understanding of how these functions behave and how they can be used to model real-world phenomena. The ability to manipulate exponential equations, solve for unknown constants, and evaluate the function at specific points is a valuable skill in various fields, from mathematics and science to finance and engineering. This problem serves as a testament to the importance of a methodical approach to problem-solving. By breaking down the problem into smaller, manageable steps, we were able to navigate the challenges and arrive at the correct solution. The strategic use of algebraic techniques, such as elimination and substitution, played a crucial role in simplifying the equations and isolating the unknowns. The experience gained from solving this problem extends beyond the specific context of exponential functions. The problem-solving skills honed here can be applied to a wide range of mathematical and scientific challenges. The ability to translate a problem into a mathematical form, devise a solution strategy, and execute the steps with precision is a hallmark of a proficient problem solver. As we conclude this exploration of exponential functions, we carry with us not only the solution to this particular problem but also a deeper understanding of the mathematical principles involved. This knowledge empowers us to tackle future challenges with confidence and competence, ready to apply our problem-solving skills to new and exciting contexts. Thus, let us continue our mathematical journey, building upon the foundations we've laid and exploring the vast landscape of mathematical ideas that await us.