Exploring Exponential And Logarithmic Functions Domain Analysis

This article delves into the fascinating world of exponential and logarithmic functions, focusing on understanding their domains and behavior. We will analyze three specific functions, namely ${ f(x) = 2^{2x+1} + 2^{x+2} }$, ${ g(x) = 3 - 2^x }$, and ${ h(x) = \log_4(x-1) - \log_4(x+2)^{-1} }$. Our primary goal is to determine the domains of these functions and explore the mathematical principles that govern them. Understanding the domain of a function is crucial in mathematics as it defines the set of input values for which the function produces a valid output. In essence, the domain provides the boundaries within which the function operates meaningfully. We will systematically examine each function, applying the relevant definitions and properties of exponential and logarithmic functions to precisely identify their respective domains. This exploration will not only enhance our understanding of these specific functions but also provide a broader insight into the nature and behavior of exponential and logarithmic functions in general. Throughout this article, we will emphasize clarity and precision, ensuring that the concepts and methodologies are accessible to a wide audience interested in mathematical analysis.

3.1 Determining the Domains of Functions f, g, and h

In this section, we embark on a detailed journey to determine the domains of the functions ${ f(x) = 2^{2x+1} + 2^{x+2} }$, ${ g(x) = 3 - 2^x }$, and ${ h(x) = \log_4(x-1) - \log_4(x+2)^{-1} }$. The domain of a function is a fundamental concept in mathematics, representing the set of all possible input values (x-values) for which the function produces a valid output. Finding the domain involves identifying any restrictions on the input values that would result in undefined or non-real outputs. These restrictions often arise from operations such as division by zero, taking the square root of a negative number, or evaluating logarithms of non-positive numbers.

Domain of f(x) = 2^(2x+1) + 2^(x+2)

Let's start with the function ${ f(x) = 2^{2x+1} + 2^{x+2} }$. This function is an exponential function, which involves a constant base raised to a variable exponent. Exponential functions are generally well-behaved and do not have the same domain restrictions as, say, rational or logarithmic functions. The key characteristic of exponential functions is that they are defined for all real numbers. There are no values of x that would cause the function to be undefined. The exponential function ${ a^x }$, where a is a positive constant, is defined for all real numbers x. In our case, we have two exponential terms, ${ 2^{2x+1} }$ and ${ 2^{x+2} }$, both of which are defined for all real numbers. Therefore, the sum of these two terms, which constitutes our function f(x), is also defined for all real numbers. We can express this mathematically by saying that the domain of f(x), denoted as ${ D_f }$, is the set of all real numbers. This is often written in interval notation as ${ (-\infty, \infty) }$.

In summary, when dealing with exponential functions, it's crucial to recognize their inherent property of being defined across the entire real number line. This understanding simplifies the process of domain determination, as we do not need to consider any specific restrictions imposed by the exponential operation itself. The domain of ${ f(x) }$ is thus a straightforward application of this principle, leading us to the conclusion that ${ D_f = (-\infty, \infty) }$.

Domain of g(x) = 3 - 2^x

Next, we consider the function ${ g(x) = 3 - 2^x }$. This function, similar to f(x), involves an exponential term, specifically ${ 2^x }$. As we established in the previous analysis, exponential functions of the form ${ a^x }$, where a is a positive constant, are defined for all real numbers x. The constant 3 in the function g(x) is a simple additive term and does not impose any restrictions on the domain. The exponential term ${ 2^x }$ is defined for all real numbers, and therefore, the function ${ g(x) = 3 - 2^x }$ is also defined for all real numbers. This is because subtracting an exponential function from a constant does not introduce any domain restrictions.

Thus, the domain of g(x), denoted as ${ D_g }$, is the set of all real numbers, which can be written in interval notation as ${ (-\infty, \infty) }$. The absence of any restrictions on x arises from the nature of exponential functions, which are inherently defined for all real number inputs. It's important to recognize that transformations of exponential functions, such as adding a constant or reflecting the function, do not alter the domain. The domain remains the entire real number line as long as the base of the exponent is a positive constant and the exponent itself is a real-valued expression. In this case, the base is 2, and the exponent is x, both of which satisfy these conditions.

In essence, the determination of the domain of ${ g(x) }$ highlights the robust nature of exponential functions and their ability to accept any real number as an input. The simplicity of this function allows us to directly apply the fundamental property of exponential functions, leading to the conclusion that ${ D_g = (-\infty, \infty) }$.

Domain of h(x) = log₄(x-1) - log₄(x+2)^(-1)

Now, let's turn our attention to the function ${ h(x) = \log_4(x-1) - \log_4(x+2)^{-1} }$. This function involves logarithmic terms, which introduce domain restrictions. Logarithmic functions are only defined for positive arguments. The argument of a logarithm must be strictly greater than zero because logarithms are the inverse of exponential functions, and exponential functions always produce positive outputs. Therefore, we need to ensure that the expressions inside the logarithms are positive.

We have two logarithmic terms in ${ h(x) }$: ${ \log_4(x-1) }$ and ${ \log_4(x+2)^{-1} }$. For the first term, ${ \log_4(x-1) }$, we require that ${ x-1 > 0 }$. Solving this inequality gives us ${ x > 1 }$. For the second term, ${ \log_4(x+2)^{-1} }$, we first simplify the expression using the property of logarithms that ${ \log_b(a^c) = c \log_b(a) }$. Thus, ${ \log_4(x+2)^{-1} = -\log_4(x+2) }$. The argument of this logarithm is ${ x+2 }$, and we require that ${ x+2 > 0 }$. Solving this inequality gives us ${ x > -2 }$.

However, we must also consider the original form of the term, ${ \log_4(x+2)^{-1} }$. The argument here is ${ (x+2)^{-1} = \frac{1}{x+2} }$. For this to be positive, we need ${ \frac{1}{x+2} > 0 }$. This inequality holds true when ${ x+2 > 0 }$, which again leads to the condition ${ x > -2 }$. Furthermore, since we have a fraction, we must also ensure that the denominator is not zero, i.e., ${ x+2 \neq 0 }$, which means ${ x \neq -2 }$.

Now, we need to consider both conditions: ${ x > 1 }$ and ${ x > -2 }$. Since x must satisfy both inequalities, we take the more restrictive condition, which is ${ x > 1 }$. Therefore, the domain of h(x), denoted as ${ D_h }$, is the set of all real numbers greater than 1. In interval notation, this is written as ${ (1, \infty) }$.

In summary, the domain of ${ h(x) }$ is determined by the restrictions imposed by the logarithmic terms. The argument of each logarithm must be positive, leading to the inequalities ${ x > 1 }$ and ${ x > -2 }$. The intersection of these conditions gives us the domain ${ D_h = (1, \infty) }$. This careful analysis highlights the importance of considering all logarithmic terms and their respective arguments when determining the domain of a function.

In conclusion, we have successfully determined the domains of the three given functions: ${ f(x) = 2^{2x+1} + 2^{x+2} }$, ${ g(x) = 3 - 2^x }$, and ${ h(x) = \log_4(x-1) - \log_4(x+2)^{-1} }$. We found that the domains of the exponential functions f(x) and g(x) are both the set of all real numbers, denoted as ${ (-\infty, \infty) }$. This is due to the inherent property of exponential functions, which are defined for all real number inputs. On the other hand, the domain of the logarithmic function h(x) is the interval ${ (1, \infty) }$, which represents all real numbers greater than 1. This restriction arises from the requirement that the arguments of logarithmic functions must be positive. Understanding the domains of functions is a fundamental aspect of mathematical analysis, as it defines the set of input values for which the function produces valid outputs. This knowledge is crucial for further analysis of the functions, such as finding their ranges, intercepts, and asymptotes, and for applications in various fields of mathematics and science. By carefully examining the properties of exponential and logarithmic functions, we have gained valuable insights into their behavior and the constraints that govern their domains.