Range Of Exponential Function F(x) = (1/7)(9)^x A Comprehensive Guide

In the fascinating world of mathematics, functions play a pivotal role in describing relationships between variables. Among the diverse types of functions, exponential functions stand out for their unique properties and wide-ranging applications. This article delves into the realm of exponential functions, specifically focusing on the function f(x) = (1/7)(9)^x, and seeks to unravel its range. Understanding the range of a function is crucial as it unveils the set of all possible output values the function can produce. By exploring the characteristics of exponential functions and analyzing the given function, we aim to determine the range of f(x) and provide a comprehensive explanation for our findings.

At the heart of our exploration lies the understanding of exponential functions. Exponential functions are characterized by a constant base raised to a variable exponent. The general form of an exponential function is f(x) = a * b^x, where a is a non-zero constant, b is the base (a positive real number not equal to 1), and x is the variable exponent. The base b determines the growth or decay behavior of the function. If b is greater than 1, the function represents exponential growth, while if b is between 0 and 1, it represents exponential decay. The constant a acts as a vertical stretch or compression factor and also determines the y-intercept of the function. Exponential functions exhibit several key properties that are essential for determining their range. One of the most important properties is that exponential functions with a positive base b always produce positive output values. This stems from the fact that raising a positive number to any power, whether positive, negative, or zero, will always result in a positive number. Another crucial property is that exponential functions are continuous, meaning their graphs have no breaks or jumps. This continuity allows us to analyze the function's behavior over its entire domain and make informed conclusions about its range. Furthermore, exponential functions exhibit asymptotic behavior. As x approaches positive infinity, the function either grows without bound (for exponential growth) or approaches zero (for exponential decay). Similarly, as x approaches negative infinity, the function either approaches zero (for exponential growth) or grows without bound (for exponential decay). These asymptotic properties play a significant role in defining the range of the function.

Now, let's turn our attention to the specific function at hand: f(x) = (1/7)(9)^x. This function is an exponential function with a base of 9 and a constant factor of 1/7. To determine the range of this function, we need to consider its behavior as x varies across its domain. First, we observe that the base 9 is greater than 1, indicating that this is an exponential growth function. This means that as x increases, the function's value will also increase without bound. Conversely, as x decreases, the function's value will approach zero but never actually reach it. The constant factor of 1/7 acts as a vertical compression, scaling down the function's values by a factor of 1/7. However, this compression does not alter the fundamental behavior of the exponential function; it still exhibits exponential growth and approaches zero as x decreases. To further analyze the function's range, we can consider the limits of the function as x approaches positive and negative infinity. As x approaches positive infinity, (9)^x grows without bound, and (1/7)(9)^x also grows without bound. This indicates that the function can take on arbitrarily large positive values. As x approaches negative infinity, (9)^x approaches zero, and (1/7)(9)^x also approaches zero. However, since (9)^x is always positive, (1/7)(9)^x will also always be positive, meaning the function will never actually reach zero. Based on this analysis, we can conclude that the range of f(x) = (1/7)(9)^x consists of all positive real numbers. The function can take on any positive value, no matter how large or small, but it will never be zero or negative.

Based on our understanding of exponential functions and the specific analysis of f(x) = (1/7)(9)^x, we can now definitively determine its range. As we established earlier, exponential functions with a positive base always produce positive output values. In this case, the base is 9, which is greater than 1, confirming that the function exhibits exponential growth and will always be positive. The constant factor of 1/7 merely scales the function's values but does not alter its positivity. As x approaches positive infinity, the function grows without bound, indicating that it can take on arbitrarily large positive values. As x approaches negative infinity, the function approaches zero, but since (9)^x is always positive, the function will never actually reach zero. Therefore, the range of f(x) = (1/7)(9)^x encompasses all real numbers greater than 0. This can be expressed mathematically as (0, ∞), where the parentheses indicate that 0 is not included in the range. In other words, the function can take on any positive value, but it will never be zero or negative.

In conclusion, the range of the exponential function f(x) = (1/7)(9)^x is all real numbers greater than 0. This determination stems from the fundamental properties of exponential functions, specifically their positive output values and asymptotic behavior. By analyzing the function's base, constant factor, and limits as x approaches positive and negative infinity, we have established that the function can take on any positive value but will never be zero or negative. Understanding the range of a function is crucial for comprehending its behavior and applications. In this case, knowing the range of f(x) = (1/7)(9)^x allows us to predict its output values for different inputs and utilize it effectively in mathematical modeling and problem-solving. The exploration of exponential functions and their ranges provides valuable insights into the world of mathematics and its ability to describe and predict real-world phenomena.