What Is ( F + G ) ( X (f+g)(x ( F + G ) ( X ]?Given: F ( X ) = X 3 − X F(x) = X^3 - X F ( X ) = X 3 − X G ( X ) = X 3 + 2 X 2 − 10 G(x) = X^3 + 2x^2 - 10 G ( X ) = X 3 + 2 X 2 − 10 Enter Your Answer In Standard Form: ( F + G ) ( X ) = □ (f+g)(x) = \square ( F + G ) ( X ) = □

Mar 13, 2025 by ADMIN 276 views

What is $(f+g)(x)$?

Introduction to Function Addition

In mathematics, functions are a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis. When dealing with functions, it's essential to understand how to combine them using various operations. One of the most common operations is function addition, which involves adding two or more functions together. In this article, we will explore the concept of function addition and learn how to find the sum of two given functions.

Understanding the Given Functions

To begin with, we are given two functions:

$f(x) = x^3 - x$
$g(x) = x^3 + 2x^2 - 10$

These functions are defined for all real numbers $x$ . Our goal is to find the sum of these two functions, denoted as $(f+g)(x)$ .

Finding the Sum of the Functions

To find the sum of the functions, we need to add the corresponding terms of the two functions. This means that we will add the terms with the same power of $x$ .

Let's start by adding the terms with the same power of $x$ :

The terms with $x^3$ are $x^3$ and $x^3$ , which add up to $2x^3$ .
The terms with $x^2$ are $0$ and $2x^2$ , which add up to $2x^2$ .
The terms with $x$ are $-x$ and $0$ , which add up to $-x$ .
The constant terms are $0$ and $-10$ , which add up to $-10$ .

Writing the Sum in Standard Form

Now that we have added the corresponding terms, we can write the sum in standard form:

$(f+g)(x) = 2x^3 + 2x^2 - x - 10$

This is the sum of the two given functions, and it is denoted as $(f+g)(x)$ .

Conclusion

In this article, we learned how to find the sum of two given functions. We started by understanding the given functions and then added the corresponding terms to find the sum. The sum was then written in standard form. This concept is essential in mathematics, and it has numerous applications in various branches of mathematics.

Example Use Cases

The concept of function addition has numerous applications in various branches of mathematics. Here are a few example use cases:

Calculus: In calculus, function addition is used to find the derivative of a function. The derivative of a function is a measure of how the function changes as the input changes.
Algebra: In algebra, function addition is used to solve systems of equations. A system of equations is a set of equations that involve multiple variables.
Analysis: In analysis, function addition is used to study the properties of functions. For example, the sum of two functions can be used to study the convergence of a sequence of functions.

Tips and Tricks

Here are a few tips and tricks to help you master the concept of function addition:

Start by understanding the given functions: Before adding the functions, make sure you understand the given functions and their properties.
Add the corresponding terms: When adding the functions, make sure to add the corresponding terms.
Write the sum in standard form: Once you have added the functions, write the sum in standard form.

Frequently Asked Questions

Here are a few frequently asked questions about function addition:

What is function addition?: Function addition is the process of adding two or more functions together.
How do I find the sum of two functions?: To find the sum of two functions, add the corresponding terms of the two functions.
What is the standard form of a function?: The standard form of a function is a way of writing the function in a specific format, with the terms arranged in a specific order.

Conclusion

In conclusion, function addition is a fundamental concept in mathematics that has numerous applications in various branches of mathematics. By understanding how to find the sum of two functions, you can solve a wide range of problems in mathematics. Remember to start by understanding the given functions, add the corresponding terms, and write the sum in standard form. With practice and patience, you can master the concept of function addition and become proficient in solving problems in mathematics.

Introduction

In our previous article, we explored the concept of function addition and learned how to find the sum of two given functions. In this article, we will answer some of the most frequently asked questions about function addition.

Q&A

Q: What is function addition?

A: Function addition is the process of adding two or more functions together. It involves combining the terms of the two functions to form a new function.

Q: How do I find the sum of two functions?

A: To find the sum of two functions, you need to add the corresponding terms of the two functions. This means that you will add the terms with the same power of x.

Q: What is the standard form of a function?

A: The standard form of a function is a way of writing the function in a specific format, with the terms arranged in a specific order. The standard form of a function is typically written as:

$f(x) = ax^n + bx^{n-1} + cx^{n-2} + ... + k$

where a, b, c, ..., k are constants, and n is a positive integer.

Q: How do I add functions with different powers of x?

A: When adding functions with different powers of x, you need to add the terms with the same power of x. For example, if you have two functions:

$f(x) = x^2 + 3x + 2$

$g(x) = 2x^2 + 4x - 1$

You would add the terms with the same power of x as follows:

$(f+g)(x) = (x^2 + 2x^2) + (3x + 4x) + (2 - 1)$

$= 3x^2 + 7x + 1$

Q: Can I add functions with different variables?

A: No, you cannot add functions with different variables. For example, if you have two functions:

$f(x) = x^2 + 3x + 2$

$g(y) = y^2 + 4y - 1$

You cannot add these functions together because they have different variables (x and y).

Q: What is the difference between function addition and function multiplication?

A: Function addition and function multiplication are two different operations. Function addition involves combining the terms of two functions to form a new function, while function multiplication involves multiplying the terms of two functions to form a new function.

Q: Can I add functions with complex numbers?

A: Yes, you can add functions with complex numbers. For example, if you have two functions:

$f(x) = x^2 + 3x + 2$

$g(x) = 2x^2 + 4x - 1 + 3i$

You can add these functions together as follows:

$(f+g)(x) = (x^2 + 2x^2) + (3x + 4x) + (2 - 1 + 3i)$

$= 3x^2 + 7x + 1 + 3i$

Q: What is the importance of function addition in mathematics?

A: Function addition is an essential concept in mathematics that has numerous applications in various branches of mathematics. It is used to solve systems of equations, find the derivative of a function, and study the properties of functions.