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

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Understanding Exponential Functions

Before we dive into the specifics of f(x) = 49(1/7)^x, let's establish a foundational understanding of exponential functions. An exponential function generally takes the form f(x) = a * b^x, where 'a' is the initial value, 'b' is the base (a positive real number not equal to 1), and 'x' is the exponent. The base 'b' dictates whether the function represents exponential growth (if b > 1) or exponential decay (if 0 < b < 1). The initial value 'a' acts as a vertical scaling factor, affecting the function's y-intercept. When 'b' is greater than 1, the function's graph rises sharply as 'x' increases, demonstrating exponential growth. In contrast, when 'b' is between 0 and 1, the graph decays towards the x-axis as 'x' increases, exhibiting exponential decay. Grasping these fundamental principles is essential for accurately analyzing and interpreting any exponential function. Understanding the roles of 'a' and 'b' allows us to predict the function's general behavior, such as its rate of change and overall trend. Moreover, it's important to recognize that exponential functions have unique properties, such as their domain and range, which differ from other common function types like linear or quadratic functions. This foundational knowledge will pave the way for a deeper analysis of our specific function, f(x) = 49(1/7)^x, and its characteristics.

Analyzing the Given Function: f(x) = 49(1/7)^x

Now, let's focus on our specific function, f(x) = 49(1/7)^x. Comparing this to the general form, we see that a = 49 and b = 1/7. Since 0 < b < 1, this function represents exponential decay. This means that as x increases, the value of f(x) will decrease, approaching zero. The initial value, a = 49, tells us that the y-intercept of the function is 49. This serves as a starting point for understanding the function's behavior. As 'x' moves away from zero, the (1/7)^x term will exert its influence, causing the function value to diminish. It's also crucial to acknowledge that exponential functions of this form are defined for all real numbers 'x'. We can input any value for 'x', whether positive, negative, or zero, and obtain a corresponding value for f(x). This is a key aspect of exponential functions that distinguishes them from functions with restricted domains, such as square root functions or logarithmic functions. Furthermore, because the base (1/7) is a fraction between 0 and 1, the function will never actually reach zero; it will only get closer and closer to it as 'x' increases. This behavior is fundamental to understanding the function's range, which we will explore further. By dissecting the components of the function – the initial value and the base – we begin to construct a comprehensive picture of its overall nature and how it behaves.

Domain of f(x) = 49(1/7)^x

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For f(x) = 49(1/7)^x, we can input any real number for x. There are no restrictions, such as division by zero or taking the square root of a negative number, that would limit the possible x-values. Exponential functions of the form a*b^x are defined for all real numbers. You can substitute any positive, negative, or zero value for 'x', and you will obtain a valid output for f(x). This is because raising a positive number to any real power is a well-defined mathematical operation. For instance, (1/7)^-2 is simply 7^2, and (1/7)^0 is 1. The ability to handle any real number input is a fundamental characteristic of exponential functions. This characteristic distinguishes them from other types of functions, such as rational functions (which may have undefined points due to division by zero) or square root functions (which are undefined for negative inputs). Consequently, one of the correct options is that the domain of f(x) is the set of all real numbers. This aligns with the general properties of exponential functions and can be visually confirmed by examining the graph of the function, which extends infinitely to both the left and right along the x-axis. This unrestricted domain makes exponential functions particularly versatile in mathematical modeling and various real-world applications.

Range of f(x) = 49(1/7)^x

The range of a function is the set of all possible output values (y-values) that the function can produce. For f(x) = 49(1/7)^x, the output will always be a positive number. This is because 49 is positive, and (1/7)^x will always be positive for any real value of x. Even as x becomes very large and negative, (1/7)^x approaches infinity, and as x becomes very large and positive, (1/7)^x approaches zero, but never actually reaches it. The horizontal asymptote of the function is y=0. The function approaches the x-axis but never intersects it. Because of this, the range is all positive real numbers. The function will never produce a negative output or zero. It is crucial to understand the behavior of exponential functions as their input values tend toward positive and negative infinity. As 'x' moves towards negative infinity, (1/7)^x becomes increasingly large, and f(x) also grows without bound. Conversely, as 'x' moves towards positive infinity, (1/7)^x gets closer and closer to zero, causing f(x) to approach zero. This asymptotic behavior is a hallmark of exponential decay functions. Therefore, the range of f(x) is y > 0, meaning the function's output values are strictly greater than zero. This eliminates the option stating that the range is the set of all real numbers, as negative values and zero are not included in the range. Understanding the range of an exponential function is essential for accurately interpreting its behavior and applying it to real-world scenarios, such as modeling population decay or radioactive decay.

Function Behavior as x Increases

Now, let's examine how the function f(x) = 49(1/7)^x behaves as x increases. As we established earlier, this is an exponential decay function. This means that as x increases, the value of f(x) decreases. For every increase of 1 in x, the function value is multiplied by the base, which is 1/7 in this case. So, as x increases by 1, the function value is multiplied by 1/7, which means it becomes 1/7 of its previous value. This consistent multiplicative decrease is the defining characteristic of exponential decay. To illustrate this, consider the function's value at x=0, which is 49. At x=1, f(x) becomes 49 * (1/7) = 7. At x=2, f(x) becomes 7 * (1/7) = 1. This pattern demonstrates the exponential decay: the function's value diminishes rapidly as x increases. It is important to distinguish this behavior from that of exponential growth functions, where the function value increases multiplicatively as x increases. The base of the exponential function dictates whether it exhibits growth or decay. In our case, the base of 1/7, being between 0 and 1, clearly indicates exponential decay. This understanding is crucial for correctly interpreting the function's behavior and predicting its values for various inputs. Furthermore, this multiplicative decay has implications for various real-world applications, such as modeling the decay of a radioactive substance or the depreciation of an asset over time.

Identifying the Correct Options

Based on our analysis, we can now identify the three correct options describing the function f(x) = 49(1/7)^x:

  1. The domain is the set of all real numbers. This is correct because exponential functions of this form are defined for all real values of x. There are no restrictions on the input values. We can plug in any real number for x, and the function will yield a valid output.
  2. The range is y > 0. This is correct because the function's output values are always positive. The function approaches zero as x increases, but it never actually reaches zero, and it will never produce negative values.
  3. As x increases by 1, each function value is multiplied by 1/7. This accurately describes the function's decay behavior. For every unit increase in x, the function's value becomes 1/7 of its previous value. This multiplicative decrease is a key characteristic of exponential decay functions.

These three options comprehensively describe the key properties of the function f(x) = 49(1/7)^x. Understanding the domain, range, and the function's behavior as x changes provides a complete picture of its characteristics. By carefully analyzing the function's formula and relating it to the general form of exponential functions, we can confidently identify the correct statements. This process underscores the importance of foundational mathematical concepts and their application in analyzing and interpreting functions. Furthermore, it highlights the interconnectedness of different mathematical ideas, such as the domain, range, and the function's behavior, in providing a holistic understanding.

Conclusion

In conclusion, by analyzing the exponential function f(x) = 49(1/7)^x, we have determined that its domain is the set of all real numbers, its range is y > 0, and as x increases by 1, each function value is multiplied by 1/7. These three statements accurately describe the function's key properties and behavior. This exploration provides a valuable understanding of exponential decay functions and their characteristics, which are essential in various mathematical and real-world contexts. Understanding the domain and range of a function, as well as its behavior as the input variable changes, is crucial for effectively using it in modeling and problem-solving. The function f(x) = 49(1/7)^x serves as a clear example of exponential decay, and its analysis provides a strong foundation for understanding other exponential functions. This comprehensive exploration demonstrates the power of mathematical analysis in revealing the underlying properties of functions and their applications. Moreover, it emphasizes the importance of relating specific functions to their general forms to better understand their behavior and characteristics.