Finding Equivalent Exponential Functions A Detailed Solution For F(x) = 3(1.7)^(4x)

In the captivating world of mathematics, exponential functions stand as fundamental building blocks, governing phenomena ranging from compound interest to population growth. Understanding how to manipulate and express these functions in equivalent forms is a crucial skill. We are presented with the exponential function f(x) = 3(1.7)^(4x), and our mission is to identify its equivalent representation from a set of given options. This exploration will delve into the properties of exponents and how they can be leveraged to transform exponential expressions while preserving their underlying mathematical essence. The challenge lies in recognizing how the base and exponent interact and how these interactions can be rearranged to yield a function that, despite its different appearance, produces the exact same output for any given input.

Exponential functions are defined by the general form f(x) = a(b)^(cx), where 'a' represents the initial value, 'b' is the base, 'c' is a constant multiplier in the exponent, and 'x' is the independent variable. The base 'b' dictates the rate of growth or decay of the function, and the exponent 'cx' determines how this rate is applied as 'x' changes. The coefficient 'a' simply scales the entire function vertically. To find an equivalent function, we need to manipulate the expression without altering its fundamental behavior. This often involves applying the laws of exponents, which provide a powerful set of tools for rewriting exponential expressions. One key property is the power of a power rule: (b^m)n = b^(mn)*. This rule allows us to combine exponents that are raised to other exponents, effectively simplifying the expression. Another important property is that any constant raised to a variable power can be rewritten as a single base raised to that same variable power. For example, if we have an expression like (2³⁾x, this can be simplified to 8^x. By applying these principles judiciously, we can navigate the complexities of exponential functions and unveil their equivalent forms.

To decipher which option aligns with f(x) = 3(1.7)^(4x), we need to methodically dissect the given function and explore the potential transformations. The function has a base of 1.7, which is raised to the power of 4x. The entire term is then multiplied by 3. The core of the problem lies in how we can handle the 4x exponent. The goal is to rewrite the function in the form f(x) = a(B)^x, where 'B' is a single constant base. This will allow us to directly compare the transformed function with the provided options. The power of a power rule becomes our primary weapon in this endeavor. By recognizing that (1.7)^(4x) is the same as (1.7⁴⁾x, we can consolidate the exponent. We first calculate 1.7 raised to the power of 4, which gives us approximately 8.3521. This allows us to rewrite the function as f(x) = 3(8.3521)^x. Now, we have a function in the desired form, making it easier to identify the equivalent function among the choices.

Now, let's dissect the provided options and compare them to our transformed function, f(x) = 3(8.3521)^x. This systematic comparison will lead us to the correct equivalent expression. Option A presents f(x) = 25.06^x. This option lacks the crucial coefficient of 3 that our original function possesses. Without this coefficient, the function's vertical scale is different, meaning it will not produce the same output as the original function for all values of 'x'. Therefore, option A is not equivalent. Option B proposes f(x) = 3(8.35)^x. This option is strikingly similar to our transformed function. It maintains the coefficient of 3 and has a base of 8.35, which is a close approximation to the 8.3521 we calculated. This suggests that option B is indeed the equivalent function we are seeking. The slight discrepancy between 8.35 and 8.3521 is likely due to rounding during the calculation of 1.7 raised to the power of 4. Option C, f(x) = 3(8.25x), introduces a fundamentally different structure. This is not an exponential function because the variable 'x' is being multiplied by the base rather than serving as an exponent. This linear term disqualifies option C as an equivalent function. Lastly, option D offers f(x) = 487.97^x. This option suffers from the same issue as option A; it lacks the coefficient of 3. Additionally, the base of 487.97 is far too large to be equivalent to our transformed function. Thus, option D is also not a match.

Through careful examination, we have systematically eliminated options A, C, and D, leaving us with a strong contender in option B. The similarity between f(x) = 3(8.35)^x and our transformed function, f(x) = 3(8.3521)^x, is undeniable. The minor difference in the base value is attributable to rounding, a common practice when dealing with decimal approximations. The coefficient of 3 is present, ensuring that the vertical scale of the function remains consistent with the original. The exponential structure is also preserved, with 'x' serving as the exponent. All these factors converge to confirm that option B is indeed the equivalent function we were searching for.

After a thorough exploration of the given options, we arrive at a conclusive verdict. Option B, f(x) = 3(8.35)^x, stands out as the equivalent function to the original f(x) = 3(1.7)^(4x). Our journey involved applying the power of a power rule to transform the original function into a more recognizable form. This transformation allowed us to directly compare the function with the provided options and identify the one that maintained both the structural integrity and the numerical consistency. The process highlighted the importance of understanding the properties of exponents and how they can be used to manipulate exponential expressions. The coefficient of 3 played a crucial role in ensuring that the vertical scale of the function was preserved. The base value, approximately 8.35, dictated the rate of exponential growth and had to be accurately represented in the equivalent function. By meticulously analyzing each option, we were able to eliminate those that deviated from these essential characteristics.

The correct answer, option B, not only matches the coefficient and base value but also maintains the exponential structure, with 'x' as the exponent. This ensures that the function's behavior is identical to the original function for all values of 'x'. The slight difference between the calculated base (8.3521) and the base in option B (8.35) underscores the importance of recognizing the role of rounding in mathematical approximations. While absolute precision is desirable, in practical applications, approximations often suffice, and understanding their impact is critical. This exercise in identifying equivalent functions demonstrates the power of algebraic manipulation and the need for a keen eye for detail. Exponential functions, with their unique properties, play a vital role in modeling real-world phenomena. Mastering the art of transforming and recognizing equivalent forms is a valuable skill in mathematics and beyond. By confidently selecting option B, we not only solve the problem at hand but also deepen our understanding of the fascinating world of exponential functions.

In conclusion, the quest to find the equivalent function to f(x) = 3(1.7)^(4x) has been a rewarding exploration of exponential function transformations. Through the strategic application of the power of a power rule, we successfully reshaped the original function into a more readily comparable form. This transformation unveiled the critical role of the base and coefficient in defining the function's behavior. The meticulous examination of each option allowed us to eliminate those that deviated in structure or numerical values. The triumph of option B, f(x) = 3(8.35)^x, underscores the importance of precision, approximation, and a deep understanding of exponential properties.

This journey into equivalent functions highlights a fundamental aspect of mathematics: the ability to express the same concept in different forms. Just as a sentence can be rephrased without altering its meaning, a mathematical function can be rewritten while preserving its essential characteristics. This skill is not merely an academic exercise; it empowers us to approach problems from diverse perspectives and to select the representation that best suits the task at hand. Exponential functions, with their widespread applications in finance, science, and engineering, demand a thorough understanding of their transformations. By mastering techniques like the power of a power rule and recognizing the significance of coefficients and bases, we equip ourselves to tackle a wide range of mathematical challenges. The solution to this particular problem serves as a stepping stone towards a deeper appreciation of the elegance and versatility of exponential functions. The ability to confidently identify equivalent functions is a testament to our growing mathematical prowess.