Function Operations And Domains F(x) = √(x+3) - 2 And G(x) = X² - 4

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to outputs based on defined relationships. Understanding how to manipulate and combine functions is crucial for solving complex problems and gaining deeper insights into mathematical concepts. This article delves into the operations of functions, specifically focusing on the functions f(x) = √(x+3) - 2 and g(x) = x² - 4. We will explore the sum, difference, product, and quotient of these functions, along with meticulously determining the domain of each resulting function. A clear grasp of domains is essential, as it dictates the set of input values for which a function produces valid outputs, preventing mathematical inconsistencies like square roots of negative numbers or division by zero. By navigating these operations and domain considerations, we aim to illuminate the versatile nature of functions and their applications.

1. Function Definitions and Domain Fundamentals

Before we delve into the operations, let's clearly define the functions and establish the concept of a function's domain. The functions at hand are f(x) = √(x+3) - 2 and g(x) = x² - 4. The domain of a function is the set of all possible input values (x-values) for which the function will produce a valid output. For the function f(x), which involves a square root, the expression inside the square root must be greater than or equal to zero to avoid imaginary numbers. This restriction arises from the fundamental property of square roots in real number system, where the square root of a negative number is undefined. Therefore, for f(x), we have the inequality x + 3 ≥ 0, which implies x ≥ -3. Thus, the domain of f(x) is all real numbers greater than or equal to -3, often expressed in interval notation as [-3, ∞). This interval notation signifies that -3 is included in the domain, and the domain extends infinitely in the positive direction. For the function g(x) = x² - 4, there are no such restrictions. Squaring any real number and subtracting 4 will always result in a real number. Hence, the domain of g(x) is all real numbers, represented in interval notation as (-∞, ∞). This signifies that any real number can be used as an input for g(x). Understanding these individual domains is critical as we move forward to combine these functions, as the domain of the resulting function will be influenced by the domains of the original functions.

2. Sum of Functions: (f + g)(x)

The sum of two functions, denoted as (f + g)(x), is found by simply adding the expressions of the individual functions. In this case, we add f(x) = √(x+3) - 2 and g(x) = x² - 4. So, (f + g)(x) = f(x) + g(x) = (√(x+3) - 2) + (x² - 4). Combining like terms, we get (f + g)(x) = x² + √(x+3) - 6. This new function represents the combined behavior of f(x) and g(x). To determine the domain of (f + g)(x), we must consider the domains of both f(x) and g(x). The domain of the sum will be the intersection of the domains of the individual functions. Since the domain of f(x) is [-3, ∞) and the domain of g(x) is (-∞, ∞), we need to find the set of x-values that are in both domains. Graphically, this means finding the overlapping region of the two domains on the number line. The domain of g(x) includes all real numbers, so the intersection is essentially determined by the domain of f(x). Therefore, the domain of (f + g)(x) is [-3, ∞). This means that the function (f + g)(x) is only defined for x-values greater than or equal to -3. This domain restriction is primarily due to the presence of the square root term in f(x), which necessitates a non-negative value inside the radical.

3. Difference of Functions: (f - g)(x)

The difference of two functions, denoted as (f - g)(x), is obtained by subtracting the expression of the second function from the first function. In this case, we subtract g(x) = x² - 4 from f(x) = √(x+3) - 2. Therefore, (f - g)(x) = f(x) - g(x) = (√(x+3) - 2) - (x² - 4). Distributing the negative sign and combining like terms, we get (f - g)(x) = √(x+3) - 2 - x² + 4 = -x² + √(x+3) + 2. This new function, (f - g)(x), represents the difference in the outputs of the individual functions f(x) and g(x). Similar to the sum of functions, the domain of the difference of functions is determined by the intersection of the domains of the individual functions. The domain of (f - g)(x) is influenced by both the domain of f(x) and the domain of g(x). As we established earlier, the domain of f(x) is [-3, ∞), and the domain of g(x) is (-∞, ∞). The intersection of these two domains is the set of x-values that are present in both. Since g(x) is defined for all real numbers, the domain of (f - g)(x) is restricted by the domain of f(x). Thus, the domain of (f - g)(x) is [-3, ∞). This restriction is, again, due to the square root term in f(x), which requires the expression inside the square root to be non-negative.

4. Product of Functions: (f g)(x)

The product of two functions, denoted as (f g)(x) or (f * g)(x), is found by multiplying the expressions of the individual functions. Here, we multiply f(x) = √(x+3) - 2 and g(x) = x² - 4. Thus, (f g)(x) = f(x) * g(x) = (√(x+3) - 2)(x² - 4). To express this product, we distribute the terms: (f g)(x) = √(x+3)(x² - 4) - 2(x² - 4). Further simplifying, we get (f g)(x) = (x² - 4)√(x+3) - 2x² + 8. This function represents the product of the outputs of the two original functions. Determining the domain of (f g)(x) follows the same principle as the sum and difference: it's the intersection of the domains of the individual functions. The domain of the product is the set of x-values for which both f(x) and g(x) are defined. We know that the domain of f(x) is [-3, ∞) and the domain of g(x) is (-∞, ∞). Therefore, the domain of (f g)(x) is the intersection of these two intervals, which is [-3, ∞). This domain is primarily dictated by the square root in f(x), ensuring that the expression inside the radical remains non-negative.

5. Quotient of Functions: (f/g)(x)

The quotient of two functions, denoted as (f/g)(x), is found by dividing the expression of the first function by the expression of the second function. In this case, we divide f(x) = √(x+3) - 2 by g(x) = x² - 4. So, (f/g)(x) = f(x) / g(x) = (√(x+3) - 2) / (x² - 4). This new function represents the ratio of the outputs of f(x) and g(x). However, there's a crucial consideration when dealing with the quotient of functions: we must ensure that the denominator is not equal to zero. Division by zero is undefined in mathematics and would lead to an invalid result. Therefore, in addition to the domain restrictions of the individual functions, we must exclude any x-values that make g(x) = 0. First, let's factor the denominator: g(x) = x² - 4 = (x - 2)(x + 2). Setting g(x) equal to zero, we get (x - 2)(x + 2) = 0, which gives us two solutions: x = 2 and x = -2. These values must be excluded from the domain. Now, let's consider the domains of f(x) and g(x). The domain of f(x) is [-3, ∞), and the domain of g(x) is (-∞, ∞). The intersection of these domains is [-3, ∞). However, we must now remove the values x = 2 and x = -2 from this interval. Since -2 is within the interval [-3, ∞) and 2 is also within this interval, we must exclude both. Therefore, the domain of (f/g)(x) is [-3, -2) U (-2, 2) U (2, ∞). This notation indicates that the domain includes all real numbers greater than or equal to -3, excluding -2 and 2. The parentheses around -2 and 2 signify that these values are not included in the domain, while the square bracket around -3 indicates that it is included.

6. Conclusion

In summary, we have explored the fundamental operations on functions – addition, subtraction, multiplication, and division – using the specific examples of f(x) = √(x+3) - 2 and g(x) = x² - 4. For each operation, we derived the resulting function and meticulously determined its domain. The domain of a function is a critical aspect, defining the set of permissible input values that yield valid outputs. When combining functions through operations, the domain of the resulting function is generally the intersection of the domains of the individual functions, with additional considerations for division, where the denominator cannot be zero. Understanding these operations and domain restrictions is paramount for a comprehensive grasp of function behavior and manipulation in mathematics. The table below summarizes our findings:

This exploration underscores the importance of not only knowing how to perform operations on functions but also understanding the underlying principles that govern their behavior, particularly in relation to their domains.