Evaluating (f-g)(144) Given F(x) = √(x+12) And G(x) = 2 - √x
In mathematics, understanding function operations is crucial for solving various problems. This article delves into the concept of function operations, specifically focusing on the subtraction of two functions. We will explore how to evaluate the result of subtracting two functions at a given point. Let's consider the functions f(x) = √(x+12) and g(x) = 2 - √x. Our goal is to determine the value of (f-g)(144). This involves understanding the notation, applying the function definitions, and performing the necessary calculations. By the end of this discussion, you will have a clear understanding of how to perform function subtraction and evaluate the resulting function at a specific point. The process involves several steps, each of which is important for arriving at the correct answer. We will break down each step in detail to ensure clarity and comprehension. This includes understanding the domain of the functions involved, as well as the order of operations required to correctly evaluate the expression. Through this detailed exploration, you will gain confidence in your ability to tackle similar problems involving function operations. Remember, the key to success in mathematics is a solid understanding of the fundamental concepts and the ability to apply them in various contexts. Let’s embark on this mathematical journey together and unravel the intricacies of function subtraction.
Defining Function Operations
Before we dive into the specific problem, it's essential to define what we mean by function operations. Just like we can perform arithmetic operations (addition, subtraction, multiplication, division) on numbers, we can also perform these operations on functions. When we talk about (f-g)(x), we are referring to the subtraction of the function g(x) from the function f(x). This means that for any given value of x, we first evaluate f(x) and g(x) separately, and then we subtract the value of g(x) from the value of f(x). The result is a new function, which we can call (f-g)(x). This new function represents the difference between the two original functions at any given point in their domain. Understanding this concept is crucial for solving problems involving function operations. It allows us to combine functions in various ways and analyze their behavior. For example, we might want to find the points where two functions intersect, which can be done by setting their difference equal to zero. Or, we might want to find the maximum or minimum value of the difference between two functions. Function operations provide a powerful toolset for analyzing and manipulating mathematical relationships. They are used extensively in calculus, where we study the rates of change of functions, and in other areas of mathematics and science. By mastering the fundamentals of function operations, you will be well-equipped to tackle more advanced topics and applications. The key is to practice and become comfortable with the notation and the underlying concepts. Each operation has its own set of rules and properties, and it's important to understand these in order to apply them correctly.
Problem Setup: f(x) = √(x+12) and g(x) = 2 - √x
In our specific problem, we are given two functions: f(x) = √(x+12) and g(x) = 2 - √x. Let's take a closer look at each function. The function f(x) is a square root function. It takes an input x, adds 12 to it, and then takes the square root of the result. The domain of f(x) is all real numbers x such that x+12 ≥ 0, which means x ≥ -12. This is because we cannot take the square root of a negative number in the real number system. The function g(x) is also a function involving a square root. It takes an input x, takes the square root of it, and then subtracts the result from 2. The domain of g(x) is all non-negative real numbers, x ≥ 0, since we cannot take the square root of a negative number. Understanding the domains of these functions is crucial because when we perform operations on functions, the domain of the resulting function is the intersection of the domains of the original functions. In this case, the domain of (f-g)(x) will be the set of all x that are in both the domain of f(x) and the domain of g(x). This means we need to consider the restrictions imposed by both functions when we evaluate (f-g)(144). Before we proceed with the calculation, it's important to ensure that the value we are evaluating at, x = 144, is within the domain of both functions. Since 144 is greater than -12 and also greater than 0, it is indeed within the domain of both f(x) and g(x), and therefore, it is in the domain of (f-g)(x). This confirms that we can proceed with the evaluation without encountering any domain issues.
Understanding (f-g)(x)
The notation (f-g)(x) represents the function obtained by subtracting the function g(x) from the function f(x). Mathematically, we can write this as: (f-g)(x) = f(x) - g(x). This means that to find the value of (f-g)(x) for a particular x, we first evaluate f(x), then evaluate g(x), and finally subtract the value of g(x) from the value of f(x). It's important to note the order of operations here. We are subtracting the entire function g(x) from the entire function f(x). This is different from other operations, such as composition of functions, where the order of operations is also crucial but follows a different pattern. In our case, we have f(x) = √(x+12) and g(x) = 2 - √x. Therefore, we can write (f-g)(x) as: (f-g)(x) = √(x+12) - (2 - √x). Now, we have a new function, (f-g)(x), which represents the difference between the two original functions. To evaluate (f-g)(144), we will substitute x = 144 into this expression and simplify. This process involves substituting the value into both square root expressions and then performing the subtraction. Understanding this step is crucial for solving the problem correctly. We are not just subtracting the expressions as they are written; we are evaluating each function at the given point and then subtracting the resulting values. This careful approach ensures that we follow the correct mathematical procedure and arrive at the accurate answer. The concept of function subtraction is a fundamental building block for more advanced mathematical concepts, such as finding the difference quotient, which is used in calculus to define the derivative of a function.
Evaluating f(144)
Now, let's evaluate f(144). Recall that f(x) = √(x+12). To find f(144), we substitute x = 144 into the expression for f(x): f(144) = √(144+12). First, we perform the addition inside the square root: 144 + 12 = 156. So, we have f(144) = √156. Now, we need to simplify the square root of 156. We can look for perfect square factors of 156. The prime factorization of 156 is 2 × 2 × 3 × 13, which can be written as 2² × 3 × 13. Thus, we can rewrite √156 as √(2² × 3 × 13). Using the property of square roots that √(ab) = √a × √b, we can separate the perfect square factor: √156 = √(2²) × √(3 × 13) = 2√39. Therefore, f(144) = 2√39. This simplified form is important because it allows us to work with a more manageable expression. While we could use a calculator to find a decimal approximation of √156, keeping the expression in its simplified radical form allows for more precise calculations and a better understanding of the value. This step demonstrates the importance of simplifying expressions whenever possible in mathematics. It not only makes calculations easier but also reveals the underlying structure of the mathematical objects we are working with. In this case, simplifying the square root allows us to see that f(144) is a multiple of √39, which may be useful in further calculations or analysis. Now that we have evaluated f(144), we can move on to evaluating g(144).
Evaluating g(144)
Next, let's evaluate g(144). Recall that g(x) = 2 - √x. To find g(144), we substitute x = 144 into the expression for g(x): g(144) = 2 - √144. We know that the square root of 144 is 12, since 12 × 12 = 144. So, we have g(144) = 2 - 12. Now, we perform the subtraction: 2 - 12 = -10. Therefore, g(144) = -10. This evaluation is straightforward and involves recognizing the perfect square of 144. It's important to remember the order of operations here. We first evaluate the square root and then perform the subtraction. This ensures that we arrive at the correct result. The value of g(144) being a negative number is perfectly acceptable, as the function g(x) is defined to allow for negative values. This highlights the importance of understanding the properties of functions and the range of values they can produce. In this case, the range of g(x) includes negative numbers because of the subtraction of the square root from 2. Now that we have evaluated both f(144) and g(144), we have all the necessary pieces to calculate (f-g)(144). The next step is to subtract the value of g(144) from the value of f(144), which will give us the final answer. This process involves careful substitution and attention to the signs of the numbers involved. By breaking down the problem into smaller steps, we have made the calculation more manageable and reduced the likelihood of errors.
Calculating (f-g)(144)
Now that we have found f(144) = 2√39 and g(144) = -10, we can calculate (f-g)(144). Recall that (f-g)(x) = f(x) - g(x). Therefore, (f-g)(144) = f(144) - g(144). Substituting the values we found, we get: (f-g)(144) = 2√39 - (-10). Remember that subtracting a negative number is the same as adding its positive counterpart. So, we have: (f-g)(144) = 2√39 + 10. This is the value of (f-g)(144). We can leave the answer in this form, as it is the exact value. If we needed a decimal approximation, we could use a calculator to find the approximate value of √39 and then perform the calculations. However, leaving the answer in its exact form, 2√39 + 10, is often preferred in mathematics, as it avoids rounding errors and provides a more precise representation of the value. This final step brings together all the previous steps and demonstrates the importance of careful calculation and attention to detail. By breaking the problem down into smaller parts, we were able to evaluate each function separately and then combine the results to find the final answer. This approach is a valuable strategy for solving complex mathematical problems. It allows us to focus on one step at a time and avoid being overwhelmed by the overall problem. The result, (f-g)(144) = 2√39 + 10, represents the difference between the two functions at the specific point x = 144. This value can be further interpreted in the context of the original functions and their applications.
Final Answer
Therefore, the value of (f-g)(144) is 2√39 + 10. This concludes our exploration of function subtraction and evaluation. We started by defining function operations, then set up the specific problem with f(x) = √(x+12) and g(x) = 2 - √x. We understood the notation (f-g)(x) and how it represents the subtraction of two functions. We then evaluated f(144) and g(144) separately, carefully simplifying the square root expression for f(144). Finally, we calculated (f-g)(144) by subtracting g(144) from f(144), arriving at the answer 2√39 + 10. This process demonstrates a systematic approach to solving problems involving function operations. By breaking down the problem into smaller, manageable steps, we can avoid errors and gain a deeper understanding of the underlying concepts. The ability to perform function operations is a fundamental skill in mathematics, and this example provides a clear illustration of how to apply this skill in a specific context. The final answer, 2√39 + 10, represents the difference in the values of the two functions at x = 144. This value can be used for further analysis or calculations, depending on the specific application. The key takeaways from this discussion are the importance of understanding function notation, the order of operations, and the process of simplifying expressions. By mastering these concepts, you will be well-equipped to tackle a wide range of mathematical problems involving functions.
