Transformation Of Exponential Functions F(x)=10^x To G(x)=10^(3x)

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In the realm of mathematics, transformations of functions play a crucial role in understanding their behavior and relationships. Specifically, transformations of exponential functions are fundamental in various applications, including modeling growth and decay phenomena. In this article, we will delve into the transformation of the exponential function f(x) = 10^x into g(x) = 10^(3x). This analysis will provide a comprehensive understanding of how the change in the function's argument affects its graph and characteristics. We will explore the specific type of transformation involved, its effect on the function's steepness and direction, and how it relates to the original function. This exploration will not only enhance your understanding of exponential functions but also provide a foundation for analyzing other types of function transformations.

Understanding the Base Function f(x) = 10^x

To truly understand the transformation, we must first grasp the essence of the base function, f(x) = 10^x. This is a classic exponential function with a base of 10. Exponential functions are characterized by their rapid growth as x increases. In this particular function, for every unit increase in x, the value of f(x) is multiplied by 10. This characteristic leads to a steep upward curve when the function is graphed.

The graph of f(x) = 10^x has several key features. It passes through the point (0, 1) because any number raised to the power of 0 is 1. As x increases, the function increases exponentially, approaching infinity. As x decreases, the function approaches 0 but never actually reaches it. This creates a horizontal asymptote at y = 0. Understanding these basic properties of the base function is crucial for identifying how transformations affect the function's behavior. The exponential growth is the core concept here, and visualizing this growth helps in predicting the changes that will occur with transformations.

Key Characteristics of f(x) = 10^x

Before we dive into the transformation, let's solidify our understanding of f(x) = 10^x by highlighting its key characteristics. This will serve as our baseline for comparison when we analyze g(x) = 10^(3x).

  1. Exponential Growth: As x increases, f(x) increases rapidly. This is the hallmark of an exponential function, where the rate of growth is proportional to the current value.
  2. Y-intercept: The graph intersects the y-axis at the point (0, 1). This is because 10^0 = 1. The y-intercept is a critical point for understanding the vertical positioning of the graph.
  3. Horizontal Asymptote: The graph approaches the x-axis (y = 0) as x decreases, but it never actually touches or crosses it. This horizontal asymptote is a result of the fact that 10^x will always be a positive number.
  4. Domain and Range: The domain of f(x) is all real numbers, meaning that x can take any value. However, the range is all positive real numbers, meaning that f(x) will always be greater than 0. The domain and range define the extent of the function's graph in the coordinate plane.

By keeping these characteristics in mind, we can better understand how the transformation to g(x) = 10^(3x) alters the behavior of the exponential function. The exponential nature of the function is key to these properties, and understanding these traits helps in analyzing transformations.

Introducing the Transformed Function g(x) = 10^(3x)

Now that we have a firm understanding of the base function f(x) = 10^x, let's introduce the transformed function g(x) = 10^(3x). The key difference between f(x) and g(x) lies in the exponent. In f(x), the exponent is simply x, while in g(x), the exponent is 3x. This seemingly small change has a significant impact on the function's behavior and its graph. Our goal is to identify and describe this impact, understanding how the factor of 3 in the exponent transforms the original function.

To analyze the transformation, we can consider the effect of multiplying x by 3 within the exponent. This type of transformation is known as a horizontal transformation, specifically a horizontal compression or stretch. To determine whether it's a compression or stretch, we need to consider the factor in the exponent. In this case, the factor is 3, which is greater than 1. This indicates that the transformation is a horizontal compression. A horizontal compression essentially squeezes the graph of the function horizontally towards the y-axis. The horizontal transformation caused by the factor in the exponent is the central concept here, affecting the function's shape.

The Significance of the 3 in the Exponent

The crucial element in understanding the transformation from f(x) = 10^x to g(x) = 10^(3x) is the presence of the 3 in the exponent. This single digit dictates the nature and magnitude of the transformation. Let's break down why this is the case.

When we replace x with 3x in the exponent, we are effectively changing the rate at which the function grows or decays. In f(x) = 10^x, the function's value increases by a factor of 10 for every unit increase in x. However, in g(x) = 10^(3x), the function's value increases by a factor of 10 for every 1/3 unit increase in x. This means that g(x) grows much faster than f(x). The rate of growth is significantly affected, making the transformed function steeper.

Consider a specific example to illustrate this point. Let's compare the values of f(x) and g(x) at x = 1. For f(x), f(1) = 10^1 = 10. For g(x), g(1) = 10^(31) = 10^3 = 1000*. Notice how much larger the value of g(1) is compared to f(1). This difference in values highlights the compression effect and the faster growth of g(x). The numerical comparison clearly demonstrates the impact of the factor in the exponent.

Identifying the Transformation: Horizontal Compression

As we've discussed, the transformation from f(x) = 10^x to g(x) = 10^(3x) is a horizontal compression. This means that the graph of g(x) is a compressed version of the graph of f(x), squeezed horizontally towards the y-axis. To fully understand this transformation, it's essential to distinguish it from other types of transformations, such as vertical stretches or compressions.

A horizontal compression occurs when the input x is multiplied by a factor greater than 1 within the function's argument. In this case, we are multiplying x by 3, which results in a compression. The factor by which the graph is compressed is the reciprocal of the multiplier, which is 1/3. This means that the graph of g(x) is compressed horizontally by a factor of 1/3 compared to f(x). The compression factor is crucial for quantifying the transformation.

Visualizing Horizontal Compression

To visualize this horizontal compression, imagine taking the graph of f(x) = 10^x and squeezing it horizontally towards the y-axis. Points on the graph of f(x) that are further away from the y-axis will be moved closer to the y-axis in the graph of g(x). For example, the point (1, 10) on the graph of f(x) will be transformed to the point (1/3, 10) on the graph of g(x). The y-coordinate remains the same, but the x-coordinate is compressed by a factor of 1/3. This graphical representation helps in understanding the geometric effect of the transformation.

The horizontal compression makes the graph of g(x) steeper than the graph of f(x). This is because the function values increase more rapidly as x increases. In essence, the function is