Solving Exponential Functions A Detailed Analysis Of G(x) = -3(1/2)^x

In this comprehensive guide, we will delve into the fascinating world of exponential functions, focusing on the specific function g(x) = -3(1/2)^x. Understanding exponential functions is crucial in various fields, including mathematics, physics, finance, and computer science. This article aims to provide a clear and detailed explanation of how to evaluate this function for different values of x, using tables and graphical representations to aid comprehension. We will explore the properties of exponential functions, their behavior, and practical applications. By the end of this guide, you will have a solid understanding of how to work with exponential functions and solve problems involving them.

Before we dive into the specifics of our function, g(x) = -3(1/2)^x, let's first understand the basics of exponential functions. An exponential function is a function of the form f(x) = ab^x, where a is a non-zero constant, 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 (b > 1) or exponential decay (0 < b < 1). In our case, we have a base of 1/2, indicating exponential decay.

The coefficient a in the exponential function affects the initial value and the direction of the function. When a is positive, the function lies above the x-axis, and when a is negative, the function is reflected across the x-axis. In our function, g(x) = -3(1/2)^x, the coefficient -3 indicates that the function is reflected across the x-axis and stretched vertically by a factor of 3. Understanding these basic components is crucial for analyzing and solving exponential functions effectively. We will explore how these components interact to shape the behavior of g(x).

Now, let's focus specifically on the function g(x) = -3(1/2)^x. This is an exponential decay function because the base (1/2) is between 0 and 1. The negative coefficient (-3) reflects the function across the x-axis. To understand the behavior of this function, we can evaluate it for various values of x and observe the corresponding values of g(x). This process will help us visualize the function's graph and understand its properties.

To begin, we can create a table of values for x and g(x). This table will serve as a foundation for understanding how the function behaves as x changes. By calculating g(x) for different x values, we can identify trends and patterns in the function's output. This table will also be useful when we move on to graphing the function, as it provides specific points that the graph will pass through. Furthermore, analyzing the table will help us understand the concept of exponential decay, as we will see g(x) approaching zero as x increases.

To determine the values of g(x) for specific x-values, we substitute each x-value into the function g(x) = -3(1/2)^x and calculate the result. Let's consider a range of x-values, including negative, zero, and positive numbers, to get a comprehensive understanding of the function's behavior. For example, we can choose x-values such as -2, -1, 0, 1, and 2.

For x = -2:

g(-2) = -3(1/2)^(-2) = -3(2^2) = -3(4) = -12

For x = -1:

g(-1) = -3(1/2)^(-1) = -3(2^1) = -3(2) = -6

For x = 0:

g(0) = -3(1/2)^(0) = -3(1) = -3

For x = 1:

g(1) = -3(1/2)^(1) = -3(1/2) = -1.5

For x = 2:

g(2) = -3(1/2)^(2) = -3(1/4) = -0.75

We can continue this process for other x-values to create a more detailed table. This table of values will not only help us understand the numerical behavior of the function but will also be crucial for graphing the function accurately. By plotting these points on a coordinate plane, we can visualize the exponential decay and the reflection caused by the negative coefficient.

This table provides a clear picture of how g(x) changes as x varies. Notice the exponential decay: as x increases, g(x) approaches 0. The negative sign indicates that the graph is below the x-axis. This comprehensive set of values allows us to understand the function's behavior more deeply and prepares us for graphical representation.

Graphing the function g(x) = -3(1/2)^x provides a visual representation of its behavior. To graph the function, we can plot the points from the table of values on a coordinate plane and connect them with a smooth curve. The x-axis represents the input values (x), and the y-axis represents the output values (g(x)).

When plotting the points, we see that the function starts with large negative values for negative x and approaches the x-axis as x increases. This behavior is characteristic of exponential decay functions. The graph is a smooth curve that decreases rapidly at first and then gradually approaches the x-axis without ever touching it. This is because the function has a horizontal asymptote at y = 0.

The graph also illustrates the effect of the negative coefficient (-3). Without the negative sign, the function would be above the x-axis, representing a typical exponential decay. However, the negative sign reflects the function across the x-axis, resulting in the graph being below the x-axis. The vertical stretch by a factor of 3 makes the graph steeper compared to a basic exponential decay function like (1/2)^x.

Visualizing the graph is crucial for understanding the function's properties, such as its domain, range, asymptotes, and overall behavior. The graph confirms that the function is decreasing and approaches 0 as x goes to infinity, and it becomes increasingly negative as x goes to negative infinity. This graphical representation provides a holistic view of the function, complementing the numerical analysis we performed earlier.

• Horizontal Asymptote: The graph approaches the x-axis (y = 0) as x goes to infinity. This is a characteristic feature of exponential decay functions.
• Y-intercept: The graph intersects the y-axis at (0, -3), which corresponds to the value of g(0).
• Decreasing Function: The function is decreasing over its entire domain, meaning that as x increases, g(x) decreases.
• Reflection: The negative coefficient reflects the graph across the x-axis.
• Vertical Stretch: The factor of 3 stretches the graph vertically, making it steeper.

Exponential functions, including g(x) = -3(1/2)^x, have numerous practical applications in various fields. Understanding these applications can provide a broader perspective on the importance of exponential functions in real-world scenarios.

• Radioactive Decay: Exponential decay is commonly used to model the decay of radioactive substances. The half-life of a radioactive material is the time it takes for half of the material to decay, which follows an exponential decay pattern.
• Financial Investments: Exponential functions are used to model compound interest and the depreciation of assets. For example, the value of an investment that loses half its value each year can be modeled using an exponential decay function.
• Population Growth and Decline: Exponential functions can model population growth (when the base is greater than 1) or population decline (when the base is between 0 and 1). This is particularly useful in ecology and demography.
• Drug Metabolism: The concentration of a drug in the bloodstream often decreases exponentially over time as the body metabolizes it. This is crucial in pharmacology for determining drug dosages and frequencies.
• Cooling and Heating: Newton's Law of Cooling describes how the temperature of an object changes over time, which follows an exponential decay pattern. This is used in various engineering and scientific applications.

By understanding these applications, we can appreciate the versatility and importance of exponential functions in solving real-world problems. The ability to analyze and solve exponential functions like g(x) = -3(1/2)^x is a valuable skill in many disciplines.

In this comprehensive guide, we have explored the exponential function g(x) = -3(1/2)^x in detail. We began by understanding the basics of exponential functions, then analyzed the specific function, created a table of values, graphed the function, and discussed its key features. Finally, we explored the practical applications of exponential functions in various fields.

By understanding the properties and behavior of exponential functions like g(x) = -3(1/2)^x, you are better equipped to tackle problems in mathematics, science, and other disciplines. The concepts and techniques discussed in this guide provide a solid foundation for further exploration of exponential functions and their applications. Remember, practice is key to mastering these concepts. Continue to explore different exponential functions and their graphs to deepen your understanding.

To further enhance your understanding of exponential functions, consider exploring the following topics:

• Transformations of Exponential Functions: Investigate how changing the parameters a and b in the function f(x) = ab^x affects the graph and behavior of the function.
• Logarithmic Functions: Understand the inverse relationship between exponential and logarithmic functions and how they can be used to solve exponential equations.
• Applications in Calculus: Explore how derivatives and integrals can be used to analyze exponential functions and solve related problems.
• Real-World Modeling: Practice modeling real-world scenarios using exponential functions, such as population growth, radioactive decay, and compound interest.

By continuing your exploration, you will gain a deeper appreciation for the power and versatility of exponential functions in mathematics and its applications.