Exploring The Exponential Function F(x) = 3(√18)^x

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In the realm of mathematics, exponential functions hold a significant place due to their unique properties and wide-ranging applications. These functions are characterized by a constant base raised to a variable exponent, leading to rapid growth or decay depending on the base's value. In this article, we will delve into the intricacies of the exponential function f(x) = 3(√18)^x, dissecting its domain, range, initial value, and other key characteristics. By meticulously examining these aspects, we aim to provide a comprehensive understanding of this function and its behavior.

Unveiling the Domain: All Real Numbers

The domain of a function encompasses all possible input values (x-values) for which the function is defined. In the case of our exponential function, f(x) = 3(√18)^x, the domain extends to all real numbers. This means that we can substitute any real number for x, whether it's positive, negative, or zero, and the function will produce a valid output. This unrestricted nature of the domain stems from the fact that the base, √18, is a positive real number. Raising a positive real number to any real power is a well-defined operation, ensuring that the function is defined for all x-values.

To further elucidate this concept, consider the following:

  • When x is a positive number, (√18)^x represents repeated multiplication of √18, resulting in a positive real number. Multiplying this by 3, the coefficient, still yields a positive real number.
  • When x is a negative number, (√18)^x is equivalent to 1/(√18)^(-x), where -x is a positive number. This results in a positive real number as well, and multiplying by 3 maintains its positivity.
  • When x is zero, (√18)^0 equals 1, and multiplying by 3 gives us 3, a defined value.

Therefore, regardless of the real number we substitute for x, the function f(x) = 3(√18)^x will always produce a defined real number output, confirming that its domain encompasses all real numbers. This makes option A, "The domain is all real numbers," an accurate statement.

Exploring the Range: y > 0

The range of a function represents the set of all possible output values (y-values) that the function can produce. In our case, the range of f(x) = 3(√18)^x is y > 0, meaning that the function's output will always be a positive real number. This can be understood by analyzing the components of the function:

  • The base, √18, is a positive real number. Raising a positive real number to any real power will always result in a positive real number.
  • The coefficient, 3, is also a positive number. Multiplying a positive number by another positive number will invariably yield a positive number.

Therefore, the function f(x) = 3(√18)^x, being a product of a positive coefficient and a positive exponential term, will always produce positive outputs. This restricts the range to all positive real numbers, expressed as y > 0. This implies that option B, "The range is y > 3," is incorrect, as the range starts from a value infinitesimally close to zero, not 3.

Unveiling the Initial Value: 3

The initial value of a function refers to the output value when the input is zero, i.e., f(0). To determine the initial value of f(x) = 3(√18)^x, we substitute x = 0 into the function:

f(0) = 3(√18)^0

Any non-zero number raised to the power of 0 equals 1. Therefore,

f(0) = 3 * 1 = 3

This signifies that the initial value of the function is 3. When x is 0, the function's output is 3. This corresponds to the y-intercept of the graph of the function, where the graph intersects the y-axis. This makes option C, "The initial value is 3," an accurate statement, while option D, "The initial value is 9," is incorrect.

Unpacking Exponential Functions: Growth and Behavior

To deeply understand the behavior of exponential functions like f(x) = 3(√18)^x, it's crucial to dissect their growth patterns. Exponential functions exhibit a distinctive growth trajectory – they either surge upward or descend sharply depending on the base's value. In our case, the base, √18, looms larger than 1, setting the stage for exponential growth. This growth manifests as the function's output soaring rapidly as x increases.

The exponential growth characteristic arises because multiplying by a number greater than 1 repeatedly escalates the value exponentially. Envision x climbing from 0 to 1, then 2, then 3, and so forth. With every stride of x, the function's value swells by a factor of √18. This compounding escalation leads to the function's swift ascent, mirroring the hallmark of exponential growth.

Let's crystallize this with a few points:

  • At x = 0, f(x) = 3, our starting mark.
  • At x = 1, f(x) = 3√18, already a jump from 3.
  • At x = 2, f(x) = 3(√18)^2 = 3 * 18 = 54, illustrating the exponential leap.

The growth pace intensifies as x marches onward, propelling the function toward infinity. This surge contrasts with linear functions, where growth unfolds at a steadier, unchanging pace. The exponential nature endows f(x) = 3(√18)^x with its unique attributes, notably its swift climb.

In stark contrast, consider functions with bases squeezed between 0 and 1, such as g(x) = 3(1/√18)^x. These showcase exponential decay. As x grows, the function's output contracts, inching closer to zero. This decay arises from repeated multiplication by a fraction, diminishing the value step by step.

The coefficient, 3 in f(x), subtly shifts the function's graph. It stretches the graph vertically, amplifying the y-values by 3. The initial value is pegged at 3, showcasing the function's start, while the range, y > 0, underscores that the function's outputs remain snugly above the x-axis.

In essence, deciphering the exponential function f(x) = 3(√18)^x necessitates recognizing the interplay between its base, exponent, and coefficient. The base dictates growth direction, the exponent governs pace, and the coefficient calibrates magnitude and initial positioning. This nuanced dance shapes the function's domain, range, and graphical portrayal, showcasing its place within the vast landscape of mathematics.

Simplifying the Base: Connecting to Exponential Growth Principles

To truly grasp the essence of the exponential function f(x) = 3(√18)^x, simplifying the base is pivotal. The base, √18, might seem complex at first glance, but we can rewrite it in a more digestible form by factoring out perfect squares. Recall that 18 can be expressed as 9 * 2, where 9 is a perfect square. Thus, √18 can be simplified as follows:

√18 = √(9 * 2) = √9 * √2 = 3√2

Substituting this simplified form back into our function, we get:

f(x) = 3(3√2)^x

Now, the function appears in a form that clearly illustrates its exponential nature. The base, 3√2, is a constant greater than 1, indicating exponential growth. This means that as x increases, the function's value will increase exponentially. The coefficient, 3, acts as a vertical stretch, scaling the function's output by a factor of 3.

The simplification unveils the underlying structure of the exponential function. It underscores the pivotal role of the base in dictating the function's growth. When the base exceeds 1, the function blossoms exponentially, while bases between 0 and 1 usher in exponential decay. The coefficient choreographs the magnitude, scaling the function vertically and setting the stage for its starting point.

Visualizing this simplified form also aids in grasping the function's graphical representation. The graph of f(x) = 3(3√2)^x will mirror the trajectory of a classical exponential growth curve, ascending ever more swiftly as x advances. The y-intercept will stand tall at 3, marking the function's inception, while the absence of x-intercepts underscores that the function never descends to the x-axis.

Furthermore, the simplified base facilitates comparisons with other exponential functions. Suppose we encounter g(x) = 5(2^x). By dissecting the bases, we can deduce that f(x) will exhibit swifter growth due to its larger base (3√2 ≈ 4.24) in contrast to the base 2 of g(x). This comparative lens amplifies our comprehension of the function's behavior and position within the exponential domain.

In essence, the act of simplifying the base is a potent tool in unraveling the mysteries of exponential functions. It lays bare the underlying growth and empowers predictions about the function's choreography. Armed with this simplified lens, we can navigate the complexities of exponential functions with heightened clarity.

Accurately Describing the Function: Selecting the Correct Statements

Based on our comprehensive analysis, we can now accurately identify the statements that correctly describe the function f(x) = 3(√18)^x:

  • A. The domain is all real numbers. This statement is accurate, as we have established that the function is defined for all real numbers.
  • B. The range is y > 3. This statement is incorrect. The range is y > 0.
  • C. The initial value is 3. This statement is accurate, as we calculated f(0) = 3.
  • D. The initial value is 9. This statement is incorrect, as the initial value is 3.

Therefore, the three statements that accurately describe the function f(x) = 3(√18)^x are:

  • A. The domain is all real numbers.
  • C. The initial value is 3.

Conclusion: Grasping the Essence of Exponential Functions

In this exploration, we have meticulously examined the exponential function f(x) = 3(√18)^x, dissecting its domain, range, initial value, and overall behavior. Through this analysis, we have gained a profound understanding of the function's characteristics and its place within the broader landscape of exponential functions. By simplifying the base, we illuminated the exponential growth nature of the function, and by evaluating the initial value, we pinpointed the function's starting point. This comprehensive approach empowers us to analyze and interpret exponential functions with confidence and precision.

Exponential functions, such as the one we've scrutinized, reign supreme in diverse mathematical and scientific realms. They serve as cornerstones in modeling population swells, radioactive decay, compound interest, and a spectrum of phenomena that exhibit swift growth or decay. Mastering the nuances of exponential functions equips us with a potent toolkit for deciphering and predicting the workings of our world.