Finding (f+g)(x) Given F(x) = 3x + 1 And G(x) = X^2 - 6

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Understanding Function Addition

In mathematics, adding functions is a fundamental operation that combines two functions to create a new function. When we talk about adding functions, we're essentially adding their outputs for the same input. This article will guide you through the process of finding the sum of two functions, specifically when f(x) = 3x + 1 and g(x) = x^2 - 6. Function addition, denoted as (f+g)(x), is defined as the sum of the individual functions, that is, (f+g)(x) = f(x) + g(x). This operation is crucial in various mathematical contexts, including calculus, algebra, and real analysis. By understanding how to combine functions through addition, we gain a deeper insight into their behavior and how they interact. This concept has far-reaching applications in fields like physics, engineering, and computer science, where complex systems are often modeled using combinations of simpler functions. The ability to manipulate and understand function operations is a core skill for anyone working with mathematical models and problem-solving. The goal is to break down the process step by step, so you can confidently tackle similar problems. Understanding the underlying principles of function addition is key to mastering more advanced mathematical concepts and their applications in the real world.

Defining the Functions

Before we dive into the addition, let's clearly define the functions we're working with. We have two functions:

  • f(x) = 3x + 1
  • g(x) = x^2 - 6

The function f(x) is a linear function, meaning its graph is a straight line. It takes an input x, multiplies it by 3, and then adds 1. This type of function is simple yet fundamental, appearing in countless applications across mathematics and other disciplines. The function g(x), on the other hand, is a quadratic function. Its graph is a parabola, and it involves squaring the input x and subtracting 6. Quadratic functions are essential in modeling various phenomena, such as projectile motion and the shape of suspension cables. Understanding the characteristics of these individual functions is crucial for comprehending their behavior when combined. For instance, the linear function will contribute a linear term to the sum, while the quadratic function will contribute a quadratic term, resulting in a polynomial function. The combination of these functions highlights the diverse ways in which different types of functions can interact and create more complex expressions. Recognizing the individual components allows us to predict and analyze the resulting function more effectively. This foundational step sets the stage for a deeper exploration of function operations.

Applying the Definition of Function Addition

Now that we have our functions defined, we can apply the definition of function addition:

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

This means we simply add the expressions for f(x) and g(x) together. This is the core step in combining the functions. The simplicity of the addition operation belies its significance. By adding the function expressions directly, we're creating a new function that represents the combined behavior of the original functions. This process allows us to analyze how different function components interact and influence each other. The resulting function will inherit characteristics from both original functions, but the specific form of the sum will depend on the nature of the individual terms. For example, if both functions are polynomials, the resulting function will also be a polynomial. The degree and coefficients of the resulting polynomial will be determined by the addition of like terms. By understanding this process, we can extend our knowledge to more complex function combinations and transformations. The addition of functions is a foundational concept that opens the door to a wide range of mathematical operations and applications.

Performing the Addition

Let's substitute the expressions for f(x) and g(x) into the equation:

(f+g)(x) = (3x + 1) + (x^2 - 6).

Now, we combine like terms. This is a crucial step in simplifying the expression. It involves identifying terms with the same variable and exponent and then adding their coefficients. This process ensures that the final expression is in its most concise and understandable form. In this case, we have a constant term (1 and -6), a linear term (3x), and a quadratic term (x^2). We will add the constant terms together while the other terms remain as they are. Combining like terms is a fundamental algebraic skill that is essential for manipulating expressions and solving equations. It allows us to reduce complexity and reveal the underlying structure of mathematical relationships. In the context of function addition, this step helps us to identify the key components of the resulting function and understand how they contribute to its overall behavior. By systematically combining like terms, we can accurately determine the form of the combined function and proceed with further analysis or applications.

Simplifying the Expression

Combining the constant terms, we get:

(f+g)(x) = x^2 + 3x + 1 - 6.

(f+g)(x) = x^2 + 3x - 5.

Therefore, the sum of the functions f(x) and g(x) is the quadratic function x^2 + 3x - 5. This simplified expression represents the combined behavior of the original functions. The quadratic term (x^2) indicates that the graph of the resulting function will be a parabola. The linear term (3x) affects the position and orientation of the parabola, while the constant term (-5) determines the vertical shift of the graph. Understanding these individual components allows us to visualize and analyze the function's behavior more effectively. The process of simplification is crucial in mathematics, as it allows us to express complex relationships in a clear and concise form. By reducing the expression to its simplest terms, we can identify key features and properties that might not be immediately apparent in a more complicated form. This step also makes it easier to perform further operations on the function, such as finding its roots, determining its maximum or minimum values, or analyzing its rate of change. The ability to simplify expressions is a fundamental skill that underpins much of mathematical reasoning and problem-solving.

Final Answer

The correct answer is A. x^2 + 3x - 5. We found this by applying the definition of function addition, substituting the given functions, and simplifying the resulting expression. This process highlights the core steps involved in combining functions, and reinforces the importance of algebraic manipulation and simplification. Understanding how to add functions is a foundational concept in mathematics, and this example demonstrates a practical application of this principle. The final answer represents the combined behavior of the original functions, and its properties can be further explored using various mathematical techniques. This exercise provides a solid foundation for tackling more complex function operations and applications. By mastering these fundamental concepts, you'll be well-equipped to handle a wide range of mathematical problems and real-world scenarios that involve functions.