Inverse Of Logarithmic Function F(x)=log₀.₅x And Its Inverse F⁻¹(x)=0.5ˣ Complete Table Values

In the fascinating realm of mathematics, functions and their inverses play a pivotal role in unraveling the intricate relationships between variables. One such captivating duo is the logarithmic function and its inverse, the exponential function. In this comprehensive exploration, we will delve into the inverse of the logarithmic function f(x) = log₀.₅x, which is f⁻¹(x) = 0.5ˣ. Our primary focus will be on completing a table of values for the inverse function, thereby gaining a deeper understanding of its behavior and properties. Furthermore, we will address the critical question of determining the values of a, b, and c that complete the table for the inverse function.

Understanding Logarithmic Functions and Their Inverses

Before we embark on the journey of exploring the inverse function, it is imperative to establish a solid foundation by understanding the fundamental concepts of logarithmic functions and their inverses. A logarithmic function is essentially the inverse of an exponential function. In simpler terms, it answers the question: "To what power must we raise the base to obtain a specific value?" The general form of a logarithmic function is expressed as:

f(x) = logₐx

where a represents the base of the logarithm, and x denotes the argument. The base a must be a positive number other than 1. The inverse of a logarithmic function is the exponential function, which takes the form:

f⁻¹(x) = aˣ

The exponential function essentially reverses the operation of the logarithmic function. It asks the question: "What value do we obtain when we raise the base a to the power of x?"

Delving into the Specific Logarithmic Function f(x) = log₀.₅x

Now that we have a firm grasp of the general concepts, let us focus our attention on the specific logarithmic function presented in the prompt: f(x) = log₀.₅x. This function employs a base of 0.5, which is a unique characteristic that influences its behavior. To fully comprehend this function, we must consider its domain, range, and graphical representation.

The domain of the logarithmic function f(x) = log₀.₅x encompasses all positive real numbers. This implies that we can only input positive values for x into the function. The range, on the other hand, spans the entire set of real numbers, indicating that the function can output any real value. The graph of f(x) = log₀.₅x exhibits a decreasing trend as x increases. This is a direct consequence of the base being less than 1.

Unveiling the Inverse Function f⁻¹(x) = 0.5ˣ

The prompt explicitly states that the inverse of the logarithmic function f(x) = log₀.₅x is f⁻¹(x) = 0.5ˣ. This is an exponential function with a base of 0.5. Similar to the logarithmic function, understanding the domain, range, and graph of the inverse function is crucial.

The domain of the exponential function f⁻¹(x) = 0.5ˣ encompasses all real numbers, signifying that we can input any real value for x. The range, however, is restricted to positive real numbers. This means that the function will only output positive values. The graph of f⁻¹(x) = 0.5ˣ exhibits a decreasing trend as x increases, mirroring the behavior of its logarithmic counterpart.

Completing the Table for the Inverse Function

The heart of the prompt lies in completing the table for the inverse function f⁻¹(x) = 0.5ˣ. The table provides a set of x-values and prompts us to determine the corresponding y-values, which represent the output of the inverse function. Let us meticulously calculate the y-values for each given x-value.

To accomplish this, we will substitute each x-value into the inverse function f⁻¹(x) = 0.5ˣ and evaluate the expression. For instance, when x = -2, we have:

f⁻¹(-2) = 0.5⁻² = (1/2)⁻² = 2² = 4

Similarly, for x = -1, we get:

f⁻¹(-1) = 0.5⁻¹ = (1/2)⁻¹ = 2¹ = 2

When x = 0, the calculation is straightforward:

f⁻¹(0) = 0.5⁰ = 1

Following this pattern, we can complete the entire table by evaluating f⁻¹(x) = 0.5ˣ for each given x-value.

Determining the Values of a, b, and c

The prompt also presents a table with missing values represented by a, b, and c. Our task is to determine the numerical values that correspond to these variables. To achieve this, we will leverage the calculations we performed in the previous section.

By substituting the appropriate x-values into the inverse function f⁻¹(x) = 0.5ˣ, we can directly obtain the values of a, b, and c. For example, if a represents the y-value when x = -2, then we already know that a = 4. We can apply this same logic to determine the values of b and c.

The Completed Table and the Values of a, b, and c

After meticulously performing the calculations, we can now present the completed table for the inverse function f⁻¹(x) = 0.5ˣ:

From this table, we can directly identify the values of a, b, and c:

• a = 4
• b = 2
• c = 1

Conclusion: A Deep Dive into Inverse Functions

In this comprehensive exploration, we have successfully navigated the realm of inverse functions, specifically focusing on the inverse of the logarithmic function f(x) = log₀.₅x, which is f⁻¹(x) = 0.5ˣ. We have elucidated the fundamental concepts of logarithmic and exponential functions, emphasizing their inverse relationship. Through meticulous calculations, we have completed a table of values for the inverse function and determined the values of a, b, and c that complete the table.

This exercise has not only enhanced our understanding of inverse functions but has also highlighted the intricate connections between different mathematical concepts. By delving into the properties and behaviors of logarithmic and exponential functions, we have gained a deeper appreciation for the elegance and power of mathematical relationships. The inverse relationship between these functions provides a valuable tool for solving equations and modeling real-world phenomena.

Find the values of a, b, and c to complete the table for the inverse function f⁻¹(x) given the function f(x)=log₀.₅x and its inverse f⁻¹(x)=0.5ˣ.

Inverse of Logarithmic Function f(x)=log₀.₅x and its Inverse f⁻¹(x)=0.5ˣ Complete Table Values