If $f(x)=7x-3$ And $g(x)=x^2-4x-8$, Find $(f+g)(x$\].A. $(f+g)(x)=x^2-11x-5$ B. $(f+g)(x)=x^2-3x-5$ C. $(f+g)(x)=x^2+3x-11$ D. $(f+g)(x)=8x^2-4x-11$

In mathematics, functions are a fundamental concept that helps us describe and analyze various mathematical relationships. When dealing with functions, we often encounter the concept of function addition, which involves combining two or more functions to create a new function. In this article, we will explore the concept of function addition and how to find the sum of two given functions.

What is Function Addition?

Function addition is a mathematical operation that involves combining two or more functions to create a new function. The resulting function is called the sum of the original functions. When adding two functions, we simply add their corresponding function values for each input value.

The Formula for Function Addition

The formula for function addition is given by:

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

where $f(x)$ and $g(x)$ are the two functions being added.

Example: Finding the Sum of Two Functions

Let's consider two functions:

$f(x) = 7x - 3$

$g(x) = x^2 - 4x - 8$

We want to find the sum of these two functions, denoted by $(f+g)(x)$.

Step 1: Write Down the Functions

First, we write down the two functions:

$f(x) = 7x - 3$

$g(x) = x^2 - 4x - 8$

Step 2: Add the Functions

Next, we add the two functions by combining their corresponding function values for each input value:

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

$$(f+g)(x) = (7x - 3) + (x^2 - 4x - 8)$$

Step 3: Simplify the Expression

Now, we simplify the expression by combining like terms:

$$(f+g)(x) = 7x - 3 + x^2 - 4x - 8$$

$$(f+g)(x) = x^2 + 3x - 11$$

Therefore, the sum of the two functions is:

$$(f+g)(x) = x^2 + 3x - 11$$

Conclusion

In this article, we explored the concept of function addition and how to find the sum of two given functions. We used the formula for function addition and applied it to two specific functions to find their sum. The resulting function was then simplified to obtain the final answer.

Answer

The correct answer is:

C. $(f+g)(x)=x^2+3x-11$

Discussion

This problem requires a basic understanding of function addition and the ability to simplify algebraic expressions. The student should be able to apply the formula for function addition and combine like terms to obtain the final answer.

Tips and Variations

• To make this problem more challenging, you can add more functions or use more complex functions.
• You can also ask the student to find the difference of two functions instead of their sum.
• To make this problem easier, you can use simpler functions or provide more guidance on how to simplify the expression.

Related Topics

• Function composition
• Function multiplication
• Algebraic expressions
• Simplifying expressions

Practice Problems

1. Find the sum of the following functions:

   $f(x) = 2x + 1$

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

2. Find the difference of the following functions:

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

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

3. Find the sum of the following functions:

   $f(x) = 3x - 2$

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

Conclusion

In this article, we will answer some frequently asked questions about function addition. Whether you are a student, teacher, or simply interested in mathematics, this article will provide you with a comprehensive understanding of function addition.

Q: What is function addition?

A: Function addition is a mathematical operation that involves combining two or more functions to create a new function. The resulting function is called the sum of the original functions.

Q: How do I add two functions?

A: To add two functions, you simply add their corresponding function values for each input value. The formula for function addition is given by:

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

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

A: Function addition and function multiplication are two different mathematical operations. Function addition involves combining two or more functions to create a new function, while function multiplication involves multiplying two or more functions to create a new function.

Q: Can I add more than two functions?

A: Yes, you can add more than two functions. The formula for function addition can be extended to include any number of functions:

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

Q: How do I simplify the expression after adding two functions?

A: To simplify the expression after adding two functions, you need to combine like terms. This involves combining the coefficients of the same variables and combining the constants.

Q: What are some common mistakes to avoid when adding functions?

A: Some common mistakes to avoid when adding functions include:

• Not combining like terms
• Not simplifying the expression
• Not using the correct formula for function addition

Q: Can I use function addition to solve real-world problems?

A: Yes, function addition can be used to solve real-world problems. For example, you can use function addition to model population growth, economic systems, and other complex systems.

Q: What are some examples of function addition in real-world applications?

A: Some examples of function addition in real-world applications include:

• Modeling population growth: You can use function addition to model the growth of a population over time.
• Economic systems: You can use function addition to model the behavior of economic systems, such as supply and demand.
• Signal processing: You can use function addition to process signals in signal processing applications.

Q: How do I practice function addition?

A: To practice function addition, you can try the following:

• Start with simple functions and gradually move to more complex functions.
• Practice adding functions with different variables and coefficients.
• Use online resources and practice problems to help you improve your skills.

Conclusion

In conclusion, function addition is an important concept in mathematics that allows us to combine two or more functions to create a new function. By understanding the formula for function addition and simplifying algebraic expressions, we can find the sum of two given functions. This article has provided you with a comprehensive understanding of function addition and has answered some frequently asked questions about this topic.

Practice Problems

1. Find the sum of the following functions:

   $f(x) = 2x + 1$

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

2. Find the difference of the following functions:

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

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

3. Find the sum of the following functions:

   $f(x) = 3x - 2$

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

Additional Resources

• Khan Academy: Function Addition
• Mathway: Function Addition
• Wolfram Alpha: Function Addition

Conclusion

In conclusion, function addition is an important concept in mathematics that allows us to combine two or more functions to create a new function. By understanding the formula for function addition and simplifying algebraic expressions, we can find the sum of two given functions. This article has provided you with a comprehensive understanding of function addition and has answered some frequently asked questions about this topic.