Exponential Functions Analysis Ranges Of F(x), G(x), And H(x)

In this article, we delve into the fascinating world of exponential functions by examining three distinct functions: f(x) = -rac{6}{11}(rac{11}{2})^x, g(x) = rac{6}{11}(rac{11}{2})^{-x}, and h(x) = -rac{6}{11}(rac{11}{2})^{-x}. Our primary goal is to identify the statement that accurately describes the properties of these functions, particularly focusing on their ranges. To achieve this, we will embark on a detailed exploration of exponential functions, their characteristics, and how transformations affect their behavior. Understanding exponential functions is crucial in various fields, including mathematics, physics, finance, and computer science, as they model phenomena involving rapid growth or decay. We will begin by dissecting the general form of exponential functions and then proceed to analyze each function individually, paying close attention to the base, coefficient, and any transformations applied. This approach will allow us to determine the range of each function accurately and identify the correct statement among the given options.

Understanding the Exponential Function

Before diving into the specific functions, it's crucial to grasp the fundamental nature of exponential functions. An exponential function generally takes the form $y = ab^x$, where 'a' represents the initial value or vertical stretch/compression, and 'b' is the base, determining the rate of growth or decay. The variable 'x' is the exponent, indicating the power to which the base is raised. The base 'b' is a positive real number not equal to 1. When b > 1, the function represents exponential growth, meaning the value of y increases as x increases. Conversely, when 0 < b < 1, the function represents exponential decay, where y decreases as x increases. The coefficient 'a' plays a vital role in vertically stretching or compressing the graph and, if negative, reflects the graph across the x-axis. This reflection significantly impacts the range of the function. Exponential functions possess several key characteristics, including a horizontal asymptote, which is the horizontal line that the graph approaches but never touches. The horizontal asymptote is typically the x-axis (y = 0) for basic exponential functions. However, vertical shifts can move the asymptote up or down. The domain of an exponential function is all real numbers, meaning x can take any value. However, the range depends on the base, coefficient, and any vertical transformations. By understanding these fundamental properties, we can effectively analyze and compare the given functions.

Analyzing $f(x)=-\frac{6}{11}(\frac{11}{2})^x$

Let's start by examining the function f(x) = -rac{6}{11}(rac{11}{2})^x. This is an exponential function with a base of rac{11}{2}, which is greater than 1, indicating exponential growth. The coefficient is -rac{6}{11}, which is negative. This negativity is crucial because it signifies a reflection across the x-axis. A standard exponential growth function with a positive coefficient would have a range of $(0, ext{infinity})$. However, the negative coefficient flips the graph, resulting in a range of $(-\infty, 0)$. To further clarify, consider what happens as x becomes very large. The term (rac{11}{2})^x grows exponentially, and multiplying by -rac{6}{11} makes the function's value approach negative infinity. As x becomes very small (i.e., a large negative number), (rac{11}{2})^x approaches 0, and so does -rac{6}{11}(rac{11}{2})^x. However, the function never actually reaches 0, due to the nature of exponential decay towards the horizontal asymptote. The horizontal asymptote for this function is y = 0. Because of the reflection across the x-axis, the function exists only for negative y-values. Therefore, the range of $f(x)$ is $(-\infty, 0)$. This understanding is foundational as we move on to analyze the other functions and compare their ranges. The reflection caused by the negative coefficient is a key aspect to keep in mind.

Analyzing $g(x)=\frac{6}{11}(\frac{11}{2})^{-x}$

Next, we turn our attention to the function g(x) = rac{6}{11}(rac{11}{2})^{-x}. This function appears similar to the previous one, but the exponent is $-x$. This negative exponent indicates that the function represents exponential decay rather than growth. To see this more clearly, we can rewrite the function as g(x) = rac{6}{11}((rac{11}{2})^{-1})^x = rac{6}{11}(rac{2}{11})^x. Now, it's evident that the base is rac{2}{11}, which is between 0 and 1, confirming exponential decay. The coefficient rac{6}{11} is positive, meaning there is no reflection across the x-axis. For a standard exponential decay function with a positive coefficient, the range is $(0, \infty)$. As x increases, (rac{2}{11})^x approaches 0, and as x decreases (becomes more negative), (rac{2}{11})^x increases exponentially. However, the function never reaches 0, again due to the horizontal asymptote at y = 0. Since the coefficient is positive, the function exists only for positive y-values. Therefore, the range of $g(x)$ is $(0, \infty)$. It is important to note the difference between the ranges of $f(x)$ and $g(x)$. The negative coefficient in $f(x)$ caused a reflection, leading to a negative range, while the positive coefficient in $g(x)$ results in a positive range. This difference underscores the significant impact of the coefficient on the function's behavior and range.

Analyzing $h(x)=-\frac{6}{11}(\frac{11}{2})^{-x}$

Finally, let's analyze the function h(x) = -rac{6}{11}(rac{11}{2})^{-x}. This function combines elements from both $f(x)$ and $g(x)$. Similar to $g(x)$, the exponent $-x$ indicates exponential decay. Rewriting the function as h(x) = -rac{6}{11}((rac{11}{2})^{-1})^x = -rac{6}{11}(rac{2}{11})^x, we see that the base is rac{2}{11}, confirming exponential decay. However, unlike $g(x)$, the coefficient is -rac{6}{11}, which is negative. This negative coefficient signifies a reflection across the x-axis. For a standard exponential decay function, the range would be $(0, \infty)$. But the negative coefficient reflects the graph, resulting in a range of $(-\infty, 0)$. As x increases, (rac{2}{11})^x approaches 0, and so does -rac{6}{11}(rac{2}{11})^x. As x decreases (becomes more negative), (rac{2}{11})^x increases exponentially, but the negative coefficient makes the function's value approach negative infinity. The horizontal asymptote for this function is y = 0. Because of the reflection across the x-axis, the function exists only for negative y-values. Therefore, the range of $h(x)$ is $(-\infty, 0)$. Comparing this to $g(x)$, we see the impact of the negative coefficient. While both have the same base, indicating exponential decay, the negative coefficient in $h(x)$ flips the range to negative values.

Determining the Correct Statement and Conclusion

Having analyzed the three functions $f(x)$, $g(x)$, and $h(x)$ individually, we can now accurately determine the correct statement regarding their ranges. To recap, we found that:

• f(x) = -rac{6}{11}(rac{11}{2})^x has a range of $(-\infty, 0)$.
• g(x) = rac{6}{11}(rac{11}{2})^{-x} has a range of $(0, \infty)$.
• h(x) = -rac{6}{11}(rac{11}{2})^{-x} has a range of $(-\infty, 0)$.

Based on these ranges, we can now evaluate potential statements. For example, a statement claiming that all three functions have the same range would be incorrect. Similarly, a statement asserting that all ranges are positive would also be false. The correct statement would need to accurately reflect the distinct ranges we've identified. For instance, a true statement might be: "The range of $f(x)$ and $h(x)$ is $(-\infty, 0)$, while the range of $g(x)$ is $(0, \infty)$." This accurately captures the differences in their ranges. In conclusion, understanding the impact of the base and coefficient on exponential functions is crucial for determining their ranges. The base dictates whether the function represents growth or decay, while the coefficient determines vertical stretches/compressions and reflections across the x-axis. By carefully analyzing these components, we can accurately predict and compare the behavior of different exponential functions.

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Which statement is true about the ranges of the three functions below?

$f(x)=-\frac{6}{11}(\frac{11}{2})^x$, $g(x)=\frac{6}{11}(\frac{11}{2})^{-x}$, $h(x)=-\frac{6}{11}(\frac{11}{2})^{-x}$

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Exponential Functions Analysis Ranges of f(x), g(x), and h(x)