Matching Domains To Functions A Comprehensive Guide With F(x) = Sqrt(x) And G(x) = 2x - 6

In this article, we delve into the fascinating world of function domains and how they interact when we combine functions through arithmetic operations. Our focus will be on two specific functions: $f(x) = \sqrt{x}$ and $g(x) = 2x - 6$. We will explore the domains of these individual functions and then analyze the domains of the combined functions $(f+g)(x)$, $(\frac{g}{f})(x)$, and $(\frac{f}{g})(x)$. Understanding function domains is crucial in mathematics as it defines the set of input values for which a function produces a valid output. We will meticulously examine the restrictions imposed by square roots and division, ensuring a comprehensive understanding of the resulting domains. The goal is to match each combined function with its correct domain from the given options: $(0, \infty)$, $[0,3) \cup (3, \infty)$, and $[0, \infty)$. Let's embark on this mathematical journey to unravel the intricacies of function domains.

1. Understanding the Domains of Individual Functions

Before diving into the combined functions, it's essential to understand the individual domains of $f(x)$ and $g(x)$. The domain of a function is the set of all possible input values (x-values) for which the function produces a real number output. Let's start with $f(x) = \sqrt{x}$. The square root function, denoted as $f(x) = \sqrt{x}$, is a fundamental concept in mathematics. Its domain is a crucial aspect to consider when working with this function, as it dictates the set of input values for which the function produces a real number output. The restriction on the square root function stems from the fact that we cannot take the square root of a negative number within the realm of real numbers. The square root of a negative number results in an imaginary number, which falls outside the scope of real number analysis. Therefore, the radicand, which is the expression under the square root symbol, must be greater than or equal to zero. For $f(x) = \sqrt{x}$, this means that $x$ must be greater than or equal to zero. Mathematically, we express this condition as $x \geq 0$. This inequality defines the domain of the function. In interval notation, the domain of $f(x) = \sqrt{x}$ is represented as $[0, \infty)$. This notation indicates that the domain includes all real numbers from zero to infinity, with the square bracket on the left indicating that zero is included in the domain. The concept of the domain is not just a theoretical consideration; it has practical implications in various mathematical and scientific applications. For instance, when modeling real-world phenomena using functions, the domain represents the set of physically meaningful input values. If a function models the distance traveled by an object over time, the domain would typically be restricted to non-negative values of time, as time cannot be negative in a physical context. Similarly, in many engineering and physics problems, certain input values might lead to physically impossible or undefined situations, and these values must be excluded from the domain. Understanding the domain of a function is also critical when performing operations such as composition and inversion. The domain of a composite function is influenced by the domains of both the inner and outer functions, and careful consideration must be given to ensure that the composition is well-defined. When finding the inverse of a function, the domain and range of the original function play a crucial role in determining the domain and range of the inverse function. The domain of a function is not merely a set of acceptable input values; it is an integral part of the function's definition and behavior. A thorough understanding of the domain is essential for accurate mathematical analysis and problem-solving. Therefore, when working with functions, especially those involving radicals, rational expressions, or other restrictions, always start by identifying the domain. This will help avoid common pitfalls and ensure the validity of your results. In summary, the domain of the square root function is a fundamental concept with wide-ranging implications in mathematics and its applications. By carefully considering the restrictions imposed by the square root, we can accurately determine the set of valid input values and ensure the integrity of our mathematical analysis.

Now, let's consider $g(x) = 2x - 6$. This is a linear function, and linear functions are defined for all real numbers. There are no restrictions on the input values for $g(x)$. Therefore, the domain of $g(x)$ is $(-\infty, \infty)$. Linear functions are a cornerstone of mathematical analysis and modeling, characterized by their simple yet powerful nature. Their defining feature is a constant rate of change, which translates graphically into a straight line. This simplicity makes them incredibly versatile, serving as building blocks for more complex mathematical constructs and finding applications in a wide array of fields. The domain of a linear function, the set of all possible input values for which the function is defined, is a fundamental aspect of its nature. Unlike functions with restrictions such as square roots or rational expressions, linear functions are remarkably unconstrained in their domain. They accept any real number as input, producing a corresponding real number as output. This unboundedness of the domain stems directly from the structure of the linear function's equation. A linear function is generally represented in the form $f(x) = mx + b$, where $m$ and $b$ are constants. The constant $m$ represents the slope of the line, dictating its steepness and direction, while $b$ represents the y-intercept, the point where the line crosses the vertical axis. The variable $x$ represents the input, and it can take on any real value without causing any mathematical inconsistencies or undefined operations. There are no denominators that could potentially become zero, no square roots of negative numbers to worry about, and no logarithms of non-positive values to consider. This lack of restrictions is a hallmark of linear functions and contributes to their widespread applicability. In practical terms, the unrestricted domain of linear functions means they can be used to model relationships where the input can take on any value within a given range. For example, a linear function could model the relationship between the number of hours worked and the amount earned, assuming there are no limits on the number of hours one can work. Similarly, it could represent the relationship between the temperature in Celsius and Fahrenheit, as these scales are linearly related across their entire range. However, it's important to recognize that while the mathematical domain of a linear function is all real numbers, the practical domain in a real-world application may be limited by physical or contextual constraints. For instance, if a linear function models the number of items produced by a machine, the domain would be restricted to non-negative integers, as it's impossible to produce a fraction of an item or a negative number of items. Understanding the difference between the mathematical domain and the practical domain is crucial for accurate modeling and interpretation. In conclusion, the domain of a linear function is the set of all real numbers, a characteristic that distinguishes it from many other types of functions. This unrestricted domain reflects the fundamental simplicity and versatility of linear functions, making them indispensable tools in mathematics, science, and engineering.

2. Analyzing the Domain of (f+g)(x)

The function $(f+g)(x)$ is defined as $f(x) + g(x)$. To find the domain of this combined function, we need to consider the domains of both $f(x)$ and $g(x)$. The domain of $(f+g)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$. In other words, the input values must be valid for both functions. To determine the domain of the function $(f+g)(x)$, which is defined as the sum of two individual functions, $f(x)$ and $g(x)$, we must embark on a careful and methodical analysis. The domain of a function is the set of all possible input values (x-values) for which the function produces a real number output. When we combine functions through arithmetic operations, such as addition, the domain of the resulting function is influenced by the domains of the individual functions involved. Specifically, the domain of the sum of two functions is the intersection of their individual domains. This means that the input values must be valid for both functions in order for the sum to be defined. If an input value is outside the domain of either $f(x)$ or $g(x)$, then it cannot be included in the domain of $(f+g)(x)$. This principle stems from the fundamental requirement that each function in the sum must produce a real number output. If one of the functions yields an undefined result or a complex number for a particular input, then the sum itself is undefined for that input. Therefore, the domain of $(f+g)(x)$ is the set of all real numbers that are simultaneously within the domains of both $f(x)$ and $g(x)$. To illustrate this concept, let's consider a scenario where $f(x)$ has a domain of $[a, b]$, representing a closed interval from $a$ to $b$, and $g(x)$ has a domain of $(c, \infty)$, representing an open interval from $c$ to infinity. The intersection of these two domains would be the set of all real numbers that are both greater than $c$ and within the interval $[a, b]$. This intersection could be another interval, a single point, or even an empty set, depending on the specific values of $a$, $b$, and $c$. If the intervals do not overlap at all, then the domain of $(f+g)(x)$ would be the empty set, indicating that there are no input values for which both functions are defined. In the context of our problem, where $f(x) = \sqrt{x}$ and $g(x) = 2x - 6$, we know that the domain of $f(x)$ is $[0, \infty)$ and the domain of $g(x)$ is $(-\infty, \infty)$. To find the domain of $(f+g)(x)$, we need to determine the intersection of these two intervals. The interval $[0, \infty)$ includes all non-negative real numbers, while the interval $(-\infty, \infty)$ includes all real numbers. The intersection of these two sets is the set of all non-negative real numbers, which is precisely the interval $[0, \infty)$. Therefore, the domain of $(f+g)(x)$ is $[0, \infty)$. This result aligns with our understanding of function domains and the principles of set intersection. It highlights the importance of considering the individual domains of functions when performing arithmetic operations and the need to ensure that all operations are well-defined for the resulting function. In summary, to find the domain of $(f+g)(x)$, we identify the domains of $f(x)$ and $g(x)$ and then determine their intersection, which represents the set of all input values for which both functions are defined.

We know that the domain of $f(x) = \sqrt{x}$ is $[0, \infty)$, and the domain of $g(x) = 2x - 6$ is $(-\infty, \infty)$. The intersection of these two domains is $[0, \infty)$. Therefore, the domain of $(f+g)(x)$ is $[0, \infty)$.

3. Determining the Domain of (g/f)(x)

The function $(\frac{g}{f})(x)$ is defined as $\frac{g(x)}{f(x)}$. When dealing with a quotient of functions, we need to consider two factors: the domains of the individual functions and the possibility of division by zero. The domain of $(\frac{g}{f})(x)$ will be the intersection of the domains of $g(x)$ and $f(x)$, but we must also exclude any values of $x$ for which $f(x) = 0$. To embark on the task of determining the domain of the function $(\frac{g}{f})(x)$, defined as the quotient of two individual functions, $g(x)$ divided by $f(x)$, we must employ a meticulous and comprehensive approach. This involves not only considering the domains of the individual functions involved but also accounting for the critical restriction that division by zero is undefined in mathematics. The domain of a quotient of functions is not simply the intersection of the domains of the numerator and the denominator. While the input values must certainly be valid for both $g(x)$ and $f(x)$, we must also ensure that the denominator, $f(x)$, does not equal zero for any value within the domain. This additional constraint stems from the fundamental principle that division by zero is an undefined operation in the real number system. Dividing by zero leads to mathematical inconsistencies and logical contradictions, and therefore, any input value that causes the denominator to become zero must be excluded from the domain of the quotient function. To illustrate this concept, let's consider a scenario where $g(x)$ has a domain of all real numbers, represented as $(-\infty, \infty)$, and $f(x)$ has a domain of $[0, \infty)$, which includes all non-negative real numbers. If $f(x)$ were simply the square root function, $f(x) = \sqrt{x}$, then $f(x)$ would equal zero when $x = 0$. In this case, even though 0 is within the domain of both $g(x)$ and the square root function itself, it must be excluded from the domain of $(\frac{g}{f})(x)$ because it would lead to division by zero. The domain of $(\frac{g}{f})(x)$ would then be $(0, \infty)$, an open interval that excludes 0. This example highlights the importance of carefully examining the denominator function and identifying any values that would make it equal to zero. In general, to find the domain of $(\frac{g}{f})(x)$, we first determine the domains of $g(x)$ and $f(x)$ individually. Then, we find the intersection of these two domains, representing the set of all input values that are valid for both functions. Finally, we identify any values within this intersection that would cause $f(x)$ to equal zero and exclude them from the domain. The resulting set of input values is the domain of $(\frac{g}{f})(x)$. In the specific context of our problem, where $f(x) = \sqrt{x}$ and $g(x) = 2x - 6$, we know that the domain of $g(x)$ is $(-\infty, \infty)$ and the domain of $f(x)$ is $[0, \infty)$. The intersection of these domains is $[0, \infty)$. However, we must also consider the values of $x$ for which $f(x) = \sqrt{x} = 0$. This occurs when $x = 0$. Therefore, we must exclude 0 from the domain. The final domain of $(\frac{g}{f})(x)$ is then $(0, \infty)$, which includes all positive real numbers but excludes 0. This thorough analysis ensures that we have identified all possible input values for which the quotient function is well-defined, taking into account both the domains of the individual functions and the critical restriction against division by zero. In summary, the domain of $(\frac{g}{f})(x)$ is determined by considering the intersection of the domains of $g(x)$ and $f(x)$ and then excluding any values that would make $f(x)$ equal to zero.

We have the domain of $g(x)$ as $(-\infty, \infty)$ and the domain of $f(x)$ as $[0, \infty)$. The intersection is $[0, \infty)$. Now we need to find when $f(x) = 0$. $\sqrt{x} = 0$ when $x = 0$. So, we must exclude $x = 0$ from the domain. Therefore, the domain of $(\frac{g}{f})(x)$ is $(0, \infty)$.

4. Determining the Domain of (f/g)(x)

Similar to the previous case, the function $(\frac{f}{g})(x)$ is defined as $\frac{f(x)}{g(x)}$. We need to consider the domains of $f(x)$ and $g(x)$, and we must exclude any values of $x$ for which $g(x) = 0$. The process of determining the domain of the function $(\frac{f}{g})(x)$, which is defined as the quotient of two individual functions, $f(x)$ divided by $g(x)$, necessitates a careful and systematic approach. This involves a thorough consideration of the domains of the individual functions, $f(x)$ and $g(x)$, and the crucial requirement of avoiding division by zero. Division by zero is an undefined operation in mathematics, and any value of $x$ that causes the denominator, $g(x)$, to become zero must be excluded from the domain of the quotient function. The domain of a quotient of functions is not simply the intersection of the domains of the numerator and the denominator. While the input values must indeed be valid for both $f(x)$ and $g(x)$, the restriction against division by zero imposes an additional constraint. This constraint is paramount in ensuring that the function is well-defined and produces meaningful results. To illustrate this concept, let's consider a scenario where $f(x)$ has a domain of $[0, \infty)$, representing all non-negative real numbers, and $g(x)$ has a domain of $(-\infty, \infty)$, representing all real numbers. If $g(x)$ were a linear function such as $g(x) = 2x - 6$, we would need to identify the value(s) of $x$ for which $g(x) = 0$. In this specific case, $2x - 6 = 0$ when $x = 3$. Therefore, even though 3 is within the domain of both $f(x)$ and $g(x)$ individually, it must be excluded from the domain of $(\frac{f}{g})(x)$ because it would lead to division by zero. The domain of $(\frac{f}{g})(x)$ would then be the set of all non-negative real numbers except for 3, which can be expressed in interval notation as $[0, 3) \cup (3, \infty)$. This union of intervals indicates that the domain includes all numbers from 0 up to (but not including) 3, as well as all numbers from 3 to infinity. This example underscores the importance of carefully examining the denominator function and identifying any values that would make it equal to zero. The process of determining the domain of $(\frac{f}{g})(x)$ typically involves the following steps: First, determine the domains of $f(x)$ and $g(x)$ individually. Second, find the intersection of these two domains, representing the set of all input values that are valid for both functions. Third, identify any values within this intersection that would cause $g(x)$ to equal zero. Finally, exclude these values from the domain. The resulting set of input values is the domain of $(\frac{f}{g})(x)$. In the specific context of our problem, where $f(x) = \sqrt{x}$ and $g(x) = 2x - 6$, we know that the domain of $f(x)$ is $[0, \infty)$ and the domain of $g(x)$ is $(-\infty, \infty)$. The intersection of these domains is $[0, \infty)$. Now, we need to find the values of $x$ for which $g(x) = 2x - 6 = 0$. Solving this equation, we find that $x = 3$. Therefore, we must exclude 3 from the domain. The final domain of $(\frac{f}{g})(x)$ is then $[0, 3) \cup (3, \infty)$, which includes all non-negative real numbers except for 3. This comprehensive analysis ensures that we have identified all possible input values for which the quotient function is well-defined, taking into account both the domains of the individual functions and the critical restriction against division by zero. In summary, the domain of $(\frac{f}{g})(x)$ is determined by considering the intersection of the domains of $f(x)$ and $g(x)$ and then excluding any values that would make $g(x)$ equal to zero.

We have the domain of $f(x)$ as $[0, \infty)$ and the domain of $g(x)$ as $(-\infty, \infty)$. The intersection is $[0, \infty)$. We need to find when $g(x) = 0$. $2x - 6 = 0$ when $x = 3$. So, we must exclude $x = 3$ from the domain. Therefore, the domain of $(\frac{f}{g})(x)$ is $[0, 3) \cup (3, \infty)$.

5. Matching Domains to Functions

Now we can match the domains to the correct functions:

• $(f+g)(x)$: Domain is $[0, \infty)$
• $(\frac{g}{f})(x)$: Domain is $(0, \infty)$
• $(\frac{f}{g})(x)$: Domain is $[0, 3) \cup (3, \infty)$

Therefore, the correct matching is:

• $(f+g)(x) \quad [0, \infty)$
• $(\frac{g}{f})(x) \quad (0, \infty)$
• $(\frac{f}{g})(x) \quad [0,3) \cup (3, \infty)$

This concludes our exploration of matching domains to functions. We have meticulously analyzed the domains of individual functions and their combinations, ensuring a solid understanding of the underlying mathematical principles.