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

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In this article, we delve into the characteristics of three exponential functions: f(x), g(x), and h(x). We will explore their definitions, analyze their ranges, and discuss their behavior as x varies. Understanding exponential functions is crucial in various fields, including mathematics, physics, finance, and computer science. Let's begin by defining the functions we will be examining. The functions are as follows:

f(x)=−611(112)xf(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^x

g(x)=611(112)−xg(x)=\frac{6}{11}\left(\frac{11}{2}\right)^{-x}

h(x)=−611(112)−xh(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{-x}

Understanding Exponential Functions

Before we dive into the specifics of each function, it's essential to understand the basics of exponential functions. An exponential function is a function of the form f(x)=a∗bxf(x) = a * b^x, where 'a' is a constant coefficient, 'b' is the base (a positive real number not equal to 1), and 'x' is the exponent. The base 'b' determines whether the function represents exponential growth (if b > 1) or exponential decay (if 0 < b < 1). The coefficient 'a' affects the vertical stretch or compression and any reflection over the x-axis if 'a' is negative. With these concepts in mind, let's take a closer look at our functions.

Analyzing f(x) = -6/11 * (11/2)^x

Considering the function f(x) = -6/11 * (11/2)^x, we identify the base as 11/2, which is greater than 1, indicating exponential growth. However, the coefficient -6/11 is negative. This negative sign reflects the graph of the function across the x-axis. As x increases, (11/2)^x increases exponentially, but multiplying by -6/11 makes the entire function negative and decreasing. To determine the range, we consider the possible output values of f(x). Since (11/2)^x is always positive, multiplying it by a negative coefficient makes f(x) always negative. As x approaches negative infinity, f(x) approaches 0 but never actually reaches it. As x approaches positive infinity, f(x) decreases without bound, approaching negative infinity. Therefore, the range of f(x) is (-∞, 0).

Analyzing g(x) = 6/11 * (11/2)^-x

Now, let's examine the function g(x) = 6/11 * (11/2)^-x. We can rewrite this function as g(x) = 6/11 * (2/11)^x since (11/2)^-x is the same as (2/11)^x. Here, the base is 2/11, which is between 0 and 1, indicating exponential decay. The coefficient 6/11 is positive, so there is no reflection across the x-axis. As x increases, (2/11)^x decreases exponentially, approaching 0. To determine the range of g(x), we again consider the possible output values. Since (2/11)^x is always positive, and we are multiplying by a positive coefficient, g(x) will always be positive. As x approaches positive infinity, g(x) approaches 0 but never reaches it. As x approaches negative infinity, g(x) increases without bound. Therefore, the range of g(x) is (0, ∞).

Analyzing h(x) = -6/11 * (11/2)^-x

Finally, let's analyze h(x) = -6/11 * (11/2)^-x. Similar to g(x), we can rewrite this as h(x) = -6/11 * (2/11)^x. The base is 2/11, indicating exponential decay, but the coefficient -6/11 is negative, reflecting the graph across the x-axis. As x increases, (2/11)^x decreases exponentially, approaching 0. However, due to the negative coefficient, the function h(x) will always be negative. As x approaches positive infinity, h(x) approaches 0 from the negative side. As x approaches negative infinity, h(x) decreases without bound. Thus, the range of h(x) is (-∞, 0).

Comparing the Ranges of f(x), g(x), and h(x)

After analyzing the three functions, we have determined the following ranges:

  • f(x): (-∞, 0)
  • g(x): (0, ∞)
  • h(x): (-∞, 0)

Comparing these ranges, we can see that f(x) and h(x) share the same range, which is all negative real numbers up to 0, while g(x) has a range of all positive real numbers. This difference in ranges is due to the presence or absence of the negative coefficient and the effect of the negative exponent in the functions.

Key Differences and Similarities

To summarize, here are the key differences and similarities between the functions:

  • f(x) and h(x) both have a range of (-∞, 0), meaning their output values are always negative. This is due to the negative coefficient in front of the exponential term.
  • g(x) has a range of (0, ∞), meaning its output values are always positive. This is because it has a positive coefficient and a decaying exponential term.
  • f(x) exhibits exponential decay reflected across the x-axis, while g(x) exhibits exponential decay. h(x) also shows exponential decay reflected across the x-axis.
  • The functions g(x) and h(x) share a base of 2/11, indicating exponential decay, while f(x) has a base of 11/2, indicating a form of exponential growth that is reflected across the x-axis due to the negative coefficient.

Conclusion

In conclusion, by analyzing the functions f(x), g(x), and h(x), we have gained a deeper understanding of how different components of exponential functions—such as the base and coefficient—affect their behavior and ranges. Exponential functions are powerful tools for modeling various real-world phenomena, and understanding their properties is essential for effective application. The range of a function is a fundamental aspect of its behavior, and by carefully examining the components of these functions, we can accurately determine their ranges and make informed comparisons.

This exploration highlights the importance of considering both the base and the coefficient when analyzing exponential functions. The base dictates whether the function grows or decays, while the coefficient determines the vertical stretch, compression, and reflection. Understanding these concepts allows us to predict and interpret the behavior of exponential functions in a variety of contexts.