Analyzing The Exponential Function F(x) = 49(1/7)^x Domain Range And Behavior

Jul 17, 2025 by Jeany 78 views

Exploring the Exponential Function f(x) = 49(1/7)^x A Comprehensive Analysis

In the realm of mathematics, exponential functions hold a prominent position, playing a crucial role in modeling various real-world phenomena, from population growth to radioactive decay. Understanding the characteristics and properties of these functions is paramount for both mathematical proficiency and practical applications. In this comprehensive analysis, we delve into the intricacies of the exponential function f(x) = 49(1/7)^x, meticulously examining its domain, range, and behavior as the input variable x changes. This exploration will not only solidify your understanding of exponential functions but also equip you with the skills to analyze and interpret similar functions effectively. So, let's embark on this mathematical journey and unravel the secrets hidden within this fascinating function.

Understanding the Function f(x) = 49(1/7)^x

To fully grasp the nature of the function f(x) = 49(1/7)^x, we must first dissect its components. This function is an exponential function, characterized by a constant base raised to a variable exponent. In this case, the base is (1/7), and the exponent is x. The coefficient 49 acts as a vertical stretch factor, influencing the function's overall scale. Exponential functions exhibit unique properties that distinguish them from other types of functions, such as polynomial or trigonometric functions. Their behavior is heavily influenced by the base, which dictates whether the function represents exponential growth (base greater than 1) or exponential decay (base between 0 and 1). In our case, the base (1/7) falls between 0 and 1, indicating that f(x) represents exponential decay. As x increases, the function's value decreases, approaching zero but never actually reaching it. This fundamental characteristic of exponential decay functions has significant implications in various applications, such as modeling the decay of radioactive substances or the depreciation of assets over time.

Domain Analysis

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the range of values you can plug into the function without encountering any mathematical errors, such as division by zero or taking the square root of a negative number. For the exponential function f(x) = 49(1/7)^x, we encounter no such restrictions. The exponent x can take on any real value, whether it's positive, negative, zero, or a fraction. There's no value of x that would lead to an undefined result. This is a hallmark of exponential functions; they are defined for all real numbers. Therefore, the domain of f(x) = 49(1/7)^x is the set of all real numbers, which can be expressed mathematically as (-∞, ∞). This means you can input any real number into the function, and it will produce a valid output.

Range Analysis

The range of a function, on the other hand, encompasses the set of all possible output values (y-values) that the function can produce. Determining the range involves analyzing the function's behavior and identifying any limitations on its output. For the exponential function f(x) = 49(1/7)^x, we observe that the base (1/7) is a positive number less than 1, signifying exponential decay. As x increases, the function's value decreases, approaching zero. However, it never actually reaches zero because any positive number raised to any power will always be positive. Similarly, as x decreases (becomes more negative), the function's value increases exponentially, growing without bound. However, since the base is positive, the function's output will always be positive. Therefore, the range of f(x) = 49(1/7)^x is the set of all positive real numbers, which can be expressed mathematically as (0, ∞). This means the function can produce any positive real number as an output, but it will never produce zero or a negative number.

Analyzing the Function's Behavior as x Increases

To gain a deeper understanding of the function f(x) = 49(1/7)^x, let's analyze its behavior as the input variable x increases. As we've established, this function represents exponential decay due to the base (1/7) being between 0 and 1. This means that as x increases, the function's value decreases. But how does it decrease? The exponential nature of the function dictates that the decrease is not linear; rather, it's a rapid, diminishing decrease. For each unit increase in x, the function's value is multiplied by the base (1/7). This means the value is reduced to one-seventh of its previous value. For instance, if x increases by 1, the function value is multiplied by (1/7). If x increases by 2, the function value is multiplied by (1/7) * (1/7) = (1/49), and so on. This pattern highlights the essence of exponential decay: the rate of decrease slows down as x increases. The function approaches zero asymptotically, meaning it gets arbitrarily close to zero but never actually reaches it. This behavior has significant implications in modeling various real-world phenomena, such as the decay of radioactive substances, where the amount of substance decreases exponentially over time, gradually approaching zero but never fully disappearing.

Identifying the Correct Statements

Now that we have a thorough understanding of the function f(x) = 49(1/7)^x, we can confidently evaluate the given statements and identify the correct ones. Let's revisit the statements:

The domain is the set of all real numbers.
The range is the set of all real numbers.
The domain is x > 0.
The range is y > 0.
As x increases by 1, each y-value is multiplied by (1/7).

Based on our analysis, we can determine the following:

Statement 1 is correct. We established that the domain of f(x) is indeed the set of all real numbers.
Statement 2 is incorrect. The range of f(x) is the set of all positive real numbers, not all real numbers.
Statement 3 is incorrect. The domain of f(x) is all real numbers, not just those greater than 0.
Statement 4 is correct. The range of f(x) is the set of all positive real numbers, which can be expressed as y > 0.
Statement 5 is correct. As x increases by 1, each y-value is multiplied by the base, which is (1/7) in this case. This is a fundamental property of exponential functions.

Therefore, the three correct statements are:

The domain is the set of all real numbers.
The range is y > 0.
As x increases by 1, each y-value is multiplied by (1/7).

In conclusion, our in-depth analysis of the exponential function f(x) = 49(1/7)^x has revealed its key characteristics and properties. We've determined that the domain encompasses all real numbers, while the range is limited to positive real numbers. We've also observed the function's exponential decay behavior, where the value decreases by a factor of (1/7) for each unit increase in x. This comprehensive exploration has not only answered the specific question but also provided a solid foundation for understanding exponential functions in general. By dissecting the function's components and analyzing its behavior, we've gained valuable insights that can be applied to various mathematical and real-world scenarios. Exponential functions are powerful tools for modeling diverse phenomena, and a thorough understanding of their properties is essential for both mathematical proficiency and practical applications.