Understanding the Concept of Sum of Functions

Oct 13, 2025 by miodeya.com 74 views

Introduction

In mathematics, functions are used to describe relationships between variables. When we have two functions, we can combine them to create a new function. This process is called the sum of functions. In this article, we will explore how to find the sum of two functions, using the given functions $f(x) = 2x^2 + 4x - 5$ and $g(x) = 6x^3 - 2x^2 + 3$ as examples.

Understanding the Concept of Sum of Functions

The sum of two functions, denoted as $(f+g)(x)$ , is a new function that is created by adding the corresponding terms of the two functions. This means that for each value of $x$ , we add the values of $f(x)$ and $g(x)$ to get the value of $(f+g)(x)$ .

Step-by-Step Solution

To find the sum of the two functions, we will follow these steps:

Step 1: Write Down the Given Functions

The given functions are:

$f(x) = 2x^2 + 4x - 5$

$g(x) = 6x^3 - 2x^2 + 3$

Step 2: Add the Corresponding Terms

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

$(f+g)(x) = (2x^2 + 4x - 5) + (6x^3 - 2x^2 + 3)$

Step 3: Combine Like Terms

Now, we will combine the like terms. This means that we will add or subtract the terms with the same power of $x$ .

$(f+g)(x) = 6x^3 + (2x^2 - 2x^2) + (4x) + (-5 + 3)$

Step 4: Simplify the Expression

Finally, we will simplify the expression by combining the like terms.

$(f+g)(x) = 6x^3 + 0 + 4x - 2$

Step 5: Write the Final Answer

The final answer is:

$(f+g)(x) = 6x^3 + 4x - 2$

Conclusion

In this article, we have learned how to find the sum of two functions. We have used the given functions $f(x) = 2x^2 + 4x - 5$ and $g(x) = 6x^3 - 2x^2 + 3$ as examples. We have followed the steps of adding the corresponding terms, combining like terms, and simplifying the expression to find the sum of the two functions. The final answer is $(f+g)(x) = 6x^3 + 4x - 2$ .

Answer Key

The correct answer is:

A. $(f+g)(x) = 6x^3 + 4x - 2$

Discussion

This problem is a basic example of finding the sum of two functions. It requires the student to understand the concept of sum of functions and to apply the steps of adding the corresponding terms, combining like terms, and simplifying the expression. The student should be able to recognize that the sum of two functions is a new function that is created by adding the corresponding terms of the two functions.

Tips and Variations

To make this problem more challenging, the student can be asked to find the sum of three or more functions.
The student can be asked to find the sum of two functions with different variables.
The student can be asked to find the sum of two functions with different coefficients.

Real-World Applications

The concept of sum of functions has many real-world applications. For example:

In physics, the sum of two functions can be used to describe the motion of an object.
In engineering, the sum of two functions can be used to describe the behavior of a system.
In economics, the sum of two functions can be used to describe the relationship between two variables.

Conclusion

Q: What is the sum of two functions?

A: The sum of two functions is a new function that is created by adding the corresponding terms of the two functions.

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

A: To find the sum of two functions, you need to follow these steps:

Write down the given functions.
Add the corresponding terms.
Combine like terms.
Simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same power of the variable. For example, in the expression $2x^2 + 4x^2$ , the terms $2x^2$ and $4x^2$ are like terms because they have the same power of $x$ .

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 4x^2$ , you can combine the like terms by adding the coefficients: $2x^2 + 4x^2 = 6x^2$ .

Q: What is the final answer for the sum of the functions $f(x) = 2x^2 + 4x - 5$ and $g(x) = 6x^3 - 2x^2 + 3$ ?

A: The final answer is $(f+g)(x) = 6x^3 + 4x - 2$ .

Q: Can I find the sum of three or more functions?

A: Yes, you can find the sum of three or more functions by following the same steps as finding the sum of two functions.

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

A: To find the sum of two functions with different variables, you need to use the same steps as finding the sum of two functions, but you need to make sure that the variables are the same.

Q: What are some real-world applications of finding the sum of two functions?

A: Some real-world applications of finding the sum of two functions include:

In physics, the sum of two functions can be used to describe the motion of an object.
In engineering, the sum of two functions can be used to describe the behavior of a system.
In economics, the sum of two functions can be used to describe the relationship between two variables.

Q: Can I use technology to find the sum of two functions?

A: Yes, you can use technology such as calculators or computer software to find the sum of two functions.

Q: What are some common mistakes to avoid when finding the sum of two functions?

A: Some common mistakes to avoid when finding the sum of two functions include:

Not following the steps of adding the corresponding terms, combining like terms, and simplifying the expression.
Not recognizing that the sum of two functions is a new function that is created by adding the corresponding terms of the two functions.
Not using the correct notation for the sum of two functions.

Conclusion

In conclusion, finding the sum of two functions is a basic concept in mathematics that has many real-world applications. The student should be able to understand the concept of sum of functions and to apply the steps of adding the corresponding terms, combining like terms, and simplifying the expression. The student should be able to recognize that the sum of two functions is a new function that is created by adding the corresponding terms of the two functions.