Understanding Exponential Functions When 0 < B < 1 Domain And Range

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In the realm of mathematical functions, exponential functions hold a significant place due to their unique properties and widespread applications. Among these functions, a particular case arises when the base, denoted as 'b', lies between 0 and 1 (0 < b < 1). This specific scenario gives rise to exponential decay, a phenomenon that governs various natural processes and mathematical models. In this article, we will delve into the characteristics of exponential functions with bases between 0 and 1, focusing on their domain, range, and graphical representation.

Understanding Exponential Functions

At its core, an exponential function is defined as f(x) = b^x, where 'b' represents the base and 'x' is the exponent. The base 'b' is a positive real number, and the exponent 'x' can be any real number. The behavior of the exponential function is heavily influenced by the value of the base 'b'. When b > 1, the function exhibits exponential growth, where the output increases rapidly as the input increases. Conversely, when 0 < b < 1, the function demonstrates exponential decay, characterized by a rapid decrease in the output as the input increases.

Delving into 0 < b < 1

When the base 'b' falls between 0 and 1, the exponential function f(x) = b^x displays distinctive characteristics that set it apart from exponential growth functions. Let's explore these characteristics in detail:

Domain:

The domain of a function encompasses all possible input values (x-values) for which the function is defined. For exponential functions of the form f(x) = b^x, where 0 < b < 1, the domain extends across all real numbers. This means that we can input any real number as the exponent 'x', and the function will produce a valid output. There are no restrictions on the values of 'x' that can be used.

To illustrate this, consider the function f(x) = (1/2)^x. We can input any real number for 'x', such as -2, 0, 1, or even irrational numbers like π, and the function will yield a defined output. This unrestricted nature of the input values is a key characteristic of exponential functions with bases between 0 and 1.

Range:

The range of a function encompasses all possible output values (y-values) that the function can produce. For exponential functions of the form f(x) = b^x, where 0 < b < 1, the range is restricted to positive real numbers. This means that the function can only produce positive outputs, and it will never produce zero or negative values.

The reason for this positive range lies in the nature of exponentiation. When we raise a positive base 'b' (where 0 < b < 1) to any real number exponent 'x', the result will always be a positive number. As 'x' becomes increasingly large, the output of the function approaches zero, but it never actually reaches zero. Similarly, as 'x' becomes increasingly negative, the output of the function becomes increasingly large, approaching infinity.

For example, consider the function f(x) = (1/2)^x. As 'x' increases, the output decreases, but it always remains positive. When x = 0, f(x) = 1. As 'x' increases to 1, 2, 3, and so on, f(x) becomes 1/2, 1/4, 1/8, and so on, approaching zero. Conversely, as 'x' decreases to -1, -2, -3, and so on, f(x) becomes 2, 4, 8, and so on, increasing without bound.

Graphical Representation:

The graphical representation of an exponential function f(x) = b^x, where 0 < b < 1, provides a visual understanding of its behavior. The graph is a curve that starts high on the left side of the coordinate plane and gradually decreases as it moves towards the right. This downward trend reflects the exponential decay nature of the function.

Key features of the graph include:

  • Horizontal Asymptote: The graph approaches the x-axis (y = 0) as 'x' increases, but it never actually touches or crosses the x-axis. This line, y = 0, is known as a horizontal asymptote.
  • Y-intercept: The graph intersects the y-axis at the point (0, 1). This is because when x = 0, f(x) = b^0 = 1.
  • Monotonically Decreasing: The graph is always decreasing as 'x' increases. This means that the function's output values continuously decrease as the input values increase.

Analyzing the Statements

Now, let's analyze the statements provided in the original prompt in light of our understanding of exponential functions with bases between 0 and 1:

Statement 1: The domain is all real numbers.

This statement is true. As we discussed earlier, the domain of an exponential function f(x) = b^x, where 0 < b < 1, encompasses all real numbers. There are no restrictions on the input values 'x' that can be used.

Statement 2: The domain is x > 0.

This statement is false. The domain is not limited to x > 0. As explained earlier, the domain includes all real numbers, including negative numbers and zero.

Statement 3: The range is all real numbers.

This statement is false. The range of an exponential function f(x) = b^x, where 0 < b < 1, is restricted to positive real numbers. The function can only produce positive outputs and will never produce zero or negative values.

In conclusion, exponential functions of the form f(x) = b^x, where 0 < b < 1, exhibit unique characteristics that distinguish them from exponential growth functions. Their domain encompasses all real numbers, while their range is restricted to positive real numbers. The graphical representation of these functions displays a curve that decreases monotonically, approaching the x-axis as a horizontal asymptote.

By understanding these properties, we can effectively model and analyze various real-world phenomena that exhibit exponential decay, such as radioactive decay, population decline, and the cooling of objects.