Domain And Range Of F(x) = 2(3^x) A Comprehensive Guide

What are the domain and range of the function f(x) = 2(3^x)?

In mathematics, the domain of a function represents the set of all possible input values (x-values) for which the function is defined, while the range represents the set of all possible output values (y-values) that the function can produce. Determining the domain and range is crucial for understanding the behavior and characteristics of a function. For the exponential function f(x) = 2(3^x), we embark on a detailed exploration to precisely identify its domain and range, shedding light on the function's behavior across the spectrum of real numbers.

The beauty of exponential functions lies in their ability to model phenomena that exhibit rapid growth or decay. At the heart of these functions is the concept of a base raised to a variable exponent, where the base dictates the rate of change and the exponent governs the input values. To decipher the domain and range of our specific exponential function, f(x) = 2(3^x), we'll meticulously dissect its structure, unraveling the potential inputs and outputs it can handle. We'll consider how the base of 3 influences the function's growth, how the coefficient of 2 scales the outputs, and how these elements collectively shape the overall behavior of the function. This comprehensive analysis will not only reveal the domain and range but also provide a deeper understanding of the exponential function's nature.

When we talk about the domain, we're essentially asking: What are all the possible values we can plug into our function for x? For the exponential function f(x) = 2(3^x), there are no inherent restrictions on the input values. We can substitute any real number for x, whether it's positive, negative, zero, or even a fraction. There's no risk of dividing by zero, taking the square root of a negative number, or encountering any other mathematical pitfall that would limit our choices for x. This freedom stems from the fundamental nature of exponential functions – they are defined for all real numbers. This is a cornerstone characteristic of exponential functions, setting them apart from functions like square roots or logarithms, which have specific domain constraints.

Consider this: 3 raised to any power, be it a large positive number, a minuscule negative fraction, or zero, will always yield a real number result. This unwavering property allows us to confidently assert that x can roam freely across the entire real number line. The exponential component, 3^x, gracefully handles any real number thrown its way. Multiplying the result by 2, the coefficient in our function, simply scales the output without introducing any new domain restrictions. The coefficient acts as a vertical stretch or compression, influencing the range but leaving the domain untouched. The implications of this are profound, as it means we can model a vast array of real-world phenomena using exponential functions, knowing that their input domain is boundless.

Therefore, the domain of f(x) = 2(3^x) is the set of all real numbers, which we express mathematically as (-∞, ∞). This interval notation signifies that x can take on any value from negative infinity to positive infinity, encompassing the entire real number line. The absence of any boundaries on the domain underscores the versatility and widespread applicability of exponential functions in mathematical modeling and analysis.

Now, let's shift our focus to the range. This is where we explore the possible output values, the y-values, that our function f(x) = 2(3^x) can generate. Unlike the domain, the range of an exponential function often exhibits certain constraints. In our case, the exponential term 3^x plays a pivotal role in shaping the range. A critical observation is that 3^x will always be a positive number, regardless of the value of x. Even when x is a large negative number, 3^x approaches zero but never actually reaches it. This positivity stems from the fundamental properties of exponentiation: a positive base raised to any power remains positive.

The coefficient 2 in f(x) = 2(3^x) comes into play here. It acts as a vertical stretch, multiplying every output of 3^x by 2. Since 3^x is always positive, multiplying it by 2 simply scales the positive values, ensuring that the result remains positive. This transformation shifts the possible outputs upwards, further solidifying the positivity of the range. The presence of the coefficient does not introduce any negative values or alter the fundamental fact that the function's output will always be above zero. Understanding this is key to accurately defining the range of our exponential function.

As x approaches negative infinity, 3^x gets infinitesimally close to zero, and consequently, 2(3^x) also approaches zero. However, it never actually reaches zero. On the other hand, as x increases towards positive infinity, 3^x grows exponentially, and so does 2(3^x). This unbounded growth in the positive direction indicates that the function can produce arbitrarily large positive values. The combination of these behaviors shapes the range, creating a boundary at zero and extending infinitely upwards.

Therefore, the range of f(x) = 2(3^x) is the set of all positive real numbers, which we express mathematically as (0, ∞). This interval notation signifies that the output values of the function can be any number greater than zero, extending infinitely upwards. The exclusion of zero, denoted by the parenthesis, is crucial as the function approaches zero but never attains it. The range provides valuable insights into the function's behavior, highlighting its positivity and its capacity for unbounded growth.

In summary, the domain of the function f(x) = 2(3^x) is (-∞, ∞), encompassing all real numbers. The range of the function is (0, ∞), representing all positive real numbers. Understanding the domain and range provides a comprehensive picture of the function's behavior, revealing its input capabilities and output limitations. This knowledge is essential for effectively applying exponential functions in various mathematical models and real-world scenarios.

The domain, being the set of all permissible inputs, stretches across the entire number line, reflecting the exponential function's ability to handle any real number as an exponent. The range, restricted to positive real numbers, showcases the function's inherent positivity and its capacity for unbounded growth. Together, the domain and range paint a vivid portrait of f(x) = 2(3^x), empowering us to predict and interpret its behavior with precision. This understanding is not just an academic exercise; it's a gateway to harnessing the power of exponential functions in fields ranging from finance and biology to physics and computer science. The insights gleaned from analyzing domain and range transcend specific functions, forming a fundamental pillar of mathematical literacy.

The correct answer is A. domain: (-∞, ∞); range: (0, ∞). This conclusion accurately reflects the comprehensive analysis we've undertaken, solidifying our understanding of exponential functions and their domain-range characteristics. This final answer is not just a solution; it's a testament to the power of mathematical reasoning and a stepping stone towards further exploration of the fascinating world of functions.