Finding (f+g)(x) Given F(x) = X^2 + 1 And G(x) = 5 - X

In the fascinating world of mathematics, functions play a pivotal role. They act as machines, taking inputs and producing outputs based on specific rules. Often, we encounter scenarios where we need to combine two or more functions, leading to new and intriguing mathematical expressions. One such combination is the sum of functions, denoted as (f+g)(x). This article delves into the concept of function composition, focusing on the specific example where f(x) = x^2 + 1 and g(x) = 5 - x, with the goal of determining (f+g)(x). We will explore the underlying principles, provide a step-by-step solution, and discuss the broader implications of function composition in various mathematical contexts.

Defining Functions and Function Composition

Before diving into the specifics, let's establish a clear understanding of functions and how they can be combined. A function is a rule that assigns a unique output value to each input value. We typically represent functions using notations like f(x), where 'f' is the name of the function and 'x' is the input variable. The output is the value that the function produces when applied to 'x'.

Function composition involves combining two or more functions to create a new function. There are various ways to combine functions, such as addition, subtraction, multiplication, and division. In this article, we are specifically interested in the sum of functions, denoted as (f+g)(x). The sum of two functions, f(x) and g(x), is defined as:

(f+g)(x) = f(x) + g(x)

This means that to find the value of (f+g)(x) for a particular input 'x', we simply add the values of f(x) and g(x) at that input.

Understanding the concept of functions and their composition is crucial for various mathematical disciplines, including calculus, algebra, and analysis. Function composition allows us to model complex relationships by combining simpler functions, making it a powerful tool for problem-solving and mathematical modeling.

Step-by-Step Solution: Finding (f+g)(x)

Now, let's apply the concept of function composition to the given functions: f(x) = x^2 + 1 and g(x) = 5 - x. Our objective is to determine the expression for (f+g)(x).

1. Identify the Functions

The first step is to clearly identify the functions involved. In this case, we have:

• f(x) = x^2 + 1
• g(x) = 5 - x

2. Apply the Definition of (f+g)(x)

Recall that the sum of two functions is defined as (f+g)(x) = f(x) + g(x). We will substitute the expressions for f(x) and g(x) into this definition.

(f+g)(x) = (x^2 + 1) + (5 - x)

3. Simplify the Expression

Next, we simplify the expression by combining like terms. This involves removing the parentheses and grouping terms with the same power of 'x'.

(f+g)(x) = x^2 + 1 + 5 - x

Rearranging the terms, we get:

(f+g)(x) = x^2 - x + 1 + 5

Combining the constant terms, we have:

(f+g)(x) = x^2 - x + 6

4. State the Result

Therefore, the sum of the functions f(x) and g(x) is:

(f+g)(x) = x^2 - x + 6

This is the final expression for (f+g)(x). By following these steps, we have successfully determined the sum of the two given functions. This process highlights the importance of understanding the definition of function composition and applying algebraic simplification techniques to arrive at the correct result.

Analyzing the Result and Exploring Implications

Having determined that (f+g)(x) = x^2 - x + 6, it is beneficial to analyze this result and explore its implications. The resulting function, (f+g)(x), is a quadratic function, characterized by the presence of the x^2 term. This means that its graph will be a parabola, a U-shaped curve. Understanding the shape and properties of the resulting function can provide valuable insights into its behavior and applications.

1. Understanding the Quadratic Function

The quadratic function (f+g)(x) = x^2 - x + 6 has several key features:

• Parabolic Shape: As mentioned earlier, the graph of a quadratic function is a parabola. The coefficient of the x^2 term determines whether the parabola opens upwards (if the coefficient is positive) or downwards (if the coefficient is negative). In this case, the coefficient of x^2 is 1, which is positive, so the parabola opens upwards.
• Vertex: The vertex is the lowest (or highest) point on the parabola. Its x-coordinate can be found using the formula x = -b / 2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, a = 1 and b = -1, so the x-coordinate of the vertex is x = -(-1) / (2 * 1) = 1/2. To find the y-coordinate, we substitute x = 1/2 into the function: (f+g)(1/2) = (1/2)^2 - (1/2) + 6 = 1/4 - 1/2 + 6 = 23/4. Therefore, the vertex of the parabola is (1/2, 23/4).
• Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. It can be found by setting x = 0 in the function: (f+g)(0) = 0^2 - 0 + 6 = 6. So, the y-intercept is (0, 6).
• X-intercepts (Roots): The x-intercepts are the points where the parabola intersects the x-axis. They can be found by solving the quadratic equation (f+g)(x) = 0. In this case, we need to solve x^2 - x + 6 = 0. We can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Here, a = 1, b = -1, and c = 6. The discriminant (b^2 - 4ac) is (-1)^2 - 4 * 1 * 6 = 1 - 24 = -23. Since the discriminant is negative, the quadratic equation has no real roots, meaning the parabola does not intersect the x-axis.

2. Implications and Applications

The function (f+g)(x) = x^2 - x + 6, being a quadratic function, has various applications in real-world scenarios. Quadratic functions are often used to model projectile motion, the shape of satellite dishes, and the design of suspension bridges. Understanding the properties of this function, such as its vertex and intercepts, can help us analyze and predict its behavior in different contexts.

For instance, if this function represented the cost of production as a function of the number of units produced, the vertex would represent the minimum cost. If it represented the height of a projectile as a function of time, the vertex would represent the maximum height reached by the projectile. The absence of real roots indicates that the function never takes on the value of zero, which might have specific interpretations depending on the context.

In the broader context of mathematics, function composition is a fundamental concept that extends beyond simple addition. It allows us to build complex functions from simpler ones, opening up possibilities for modeling intricate relationships and solving challenging problems. The principles we've explored in this example with f(x) = x^2 + 1 and g(x) = 5 - x can be applied to various other function combinations and mathematical scenarios.

Common Mistakes and How to Avoid Them

When working with function composition, it's easy to make mistakes if one is not careful. Here are some common errors and how to avoid them:

1. Incorrectly Applying the Definition of (f+g)(x)

A common mistake is to misunderstand the definition of (f+g)(x). Remember that (f+g)(x) means f(x) + g(x), not f(x) * g(x) or some other combination. Always refer to the correct definition and substitute the expressions for f(x) and g(x) accordingly.

How to Avoid: Reinforce your understanding of the definition by practicing various examples. Write down the definition before starting the problem to ensure you have it in mind.

2. Errors in Algebraic Simplification

After substituting the expressions, algebraic simplification is crucial. Mistakes in expanding, combining like terms, or handling signs can lead to incorrect results. Common errors include forgetting to distribute a negative sign, incorrectly combining terms with different powers of 'x', or making arithmetic errors.

How to Avoid: Take your time and carefully review each step of the simplification process. Pay close attention to signs and use parentheses to avoid confusion. Break down complex expressions into smaller steps to reduce the chance of error.

3. Misinterpreting the Result

Once you have found the expression for (f+g)(x), it's important to understand what it represents. Misinterpreting the result can lead to incorrect conclusions or applications. For example, if the resulting function is a quadratic, knowing its properties (vertex, intercepts, etc.) is essential.

How to Avoid: Take the time to analyze the resulting function. Determine its type (linear, quadratic, etc.) and identify its key features. Consider the context of the problem and how the function might be applied.

4. Forgetting the Domain

While not directly relevant in this specific example, the domain of a function is an important consideration in general function composition problems. The domain of (f+g)(x) is the intersection of the domains of f(x) and g(x). Forgetting to consider the domain can lead to incorrect results in more complex scenarios.

How to Avoid: Always be mindful of the domains of the functions involved. Identify any restrictions on the input values (e.g., division by zero, square roots of negative numbers) and ensure that the resulting function is only evaluated within its valid domain.

By being aware of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and understanding of function composition.

Conclusion

In conclusion, determining the sum of functions, (f+g)(x), is a fundamental concept in mathematics with broad applications. By understanding the definition of function composition and applying algebraic simplification techniques, we can successfully find the expression for (f+g)(x). In the specific case where f(x) = x^2 + 1 and g(x) = 5 - x, we found that (f+g)(x) = x^2 - x + 6. This resulting quadratic function exhibits a parabolic shape and has specific properties that can be analyzed and interpreted in various contexts.

Function composition is a powerful tool for modeling complex relationships and solving mathematical problems. By mastering this concept, you can enhance your problem-solving skills and gain a deeper appreciation for the beauty and versatility of mathematics. Remember to practice regularly, pay attention to details, and always strive for a thorough understanding of the underlying principles.

This article has provided a comprehensive guide to understanding function composition, including a step-by-step solution, analysis of the result, discussion of common mistakes, and broader implications. We hope that this knowledge will empower you to confidently tackle function composition problems and further explore the fascinating world of mathematics.