For { F(x) = 3x + 1 $}$ And { G(x) = X^2 - 6 $}$, Find { (f+g)(x) $}$.A. { X^2 + 3x + 7 $}$B. { X^2 + 3x - 5 $}$C. { 3x^3 - 5 $}$D. { 3x^2 - 17 $}$

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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 function addition or sum. In this article, we will explore how to find the sum of two functions, using the given functions { f(x) = 3x + 1 $}$ and { g(x) = x^2 - 6 $}$ as examples.

What is Function Addition?

Function addition is a process of combining two or more functions to create a new function. The resulting function is called the sum of the original functions. When we add two functions, we add their corresponding terms. For example, if we have two functions { f(x) = 2x + 1 $}$ and { g(x) = x^2 - 3 $}$, their sum is { (f+g)(x) = 2x + 1 + x^2 - 3 = x^2 + 2x - 2 $}$.

Step 1: Identify the Functions

To find the sum of two functions, we need to identify the functions we want to add. In this case, we have two functions:

  • { f(x) = 3x + 1 $}$
  • { g(x) = x^2 - 6 $}$

Step 2: Add the Functions

To add the functions, we add their corresponding terms. We can start by adding the terms with the same variable, which in this case is { x $}$. We have { 3x $}$ from the first function and { x^2 $}$ from the second function. Since they have different variables, we cannot add them directly. However, we can add the constant terms, which are { 1 $}$ from the first function and { -6 $}$ from the second function.

Step 3: Simplify the Result

After adding the functions, we need to simplify the result. We can start by combining like terms. In this case, we have { 3x $}$ and { x^2 $}$, which cannot be combined. However, we can combine the constant terms, which are { 1 $}$ and { -6 $}$. The sum of these terms is { -5 $}$.

Step 4: Write the Final Answer

After simplifying the result, we can write the final answer. The sum of the two functions is { (f+g)(x) = x^2 + 3x - 5 $}$.

Conclusion

In this article, we explored how to find the sum of two functions. We used the given functions { f(x) = 3x + 1 $}$ and { g(x) = x^2 - 6 $}$ as examples and followed the steps to add the functions and simplify the result. The final answer is { (f+g)(x) = x^2 + 3x - 5 $}$.

Answer

The correct answer is { B. [$ x^2 + 3x - 5 $}$ $].

Example Problems

  • Find the sum of the functions { f(x) = 2x + 3 $}$ and { g(x) = x^2 - 2 $}$.
  • Find the sum of the functions { f(x) = x^2 + 1 $}$ and { g(x) = 3x - 4 $}$.

Solutions

  • The sum of the functions { f(x) = 2x + 3 $}$ and { g(x) = x^2 - 2 $}$ is { (f+g)(x) = x^2 + 2x + 1 $}$.
  • The sum of the functions { f(x) = x^2 + 1 $}$ and { g(x) = 3x - 4 $}$ is { (f+g)(x) = x^2 + 3x - 3 $}$.

Tips and Tricks

  • When adding functions, make sure to add the corresponding terms.
  • When simplifying the result, combine like terms.
  • When writing the final answer, make sure to include the variable and the constant term.

Practice Problems

  • Find the sum of the functions { f(x) = x^2 + 2 $}$ and { g(x) = 3x - 1 $}$.
  • Find the sum of the functions { f(x) = 2x + 1 $}$ and { g(x) = x^2 - 3 $}$.

Solutions

  • The sum of the functions { f(x) = x^2 + 2 $}$ and { g(x) = 3x - 1 $}$ is { (f+g)(x) = x^2 + 3x + 1 $}$.
  • The sum of the functions { f(x) = 2x + 1 $}$ and { g(x) = x^2 - 3 $}$ is { (f+g)(x) = x^2 + 2x - 2 $}$.
    Q&A: Finding the Sum of Two Functions =====================================

Q: What is function addition?

A: Function addition is a process of 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 need to add their corresponding terms. You can start by adding the terms with the same variable, and then add the constant terms.

Q: What if the functions have different variables?

A: If the functions have different variables, you cannot add them directly. However, you can add the constant terms.

Q: How do I simplify the result?

A: To simplify the result, you need to combine like terms. This means combining terms with the same variable.

Q: What is the final answer?

A: The final answer is the sum of the two functions, which is the resulting function after adding the original functions.

Q: Can I use function addition to find the difference between two functions?

A: No, function addition is used to find the sum of two functions, not the difference. To find the difference between two functions, you need to use function subtraction.

Q: How do I subtract two functions?

A: To subtract two functions, you need to subtract the corresponding terms. You can start by subtracting the terms with the same variable, and then subtract the constant terms.

Q: What if the functions have different variables?

A: If the functions have different variables, you cannot subtract them directly. However, you can subtract the constant terms.

Q: How do I simplify the result?

A: To simplify the result, you need to combine like terms. This means combining terms with the same variable.

Q: What is the final answer?

A: The final answer is the difference between the two functions, which is the resulting function after subtracting the original functions.

Q: Can I use function addition and subtraction to find the product of two functions?

A: No, function addition and subtraction are used to find the sum and difference of two functions, not the product. To find the product of two functions, you need to use function multiplication.

Q: How do I multiply two functions?

A: To multiply two functions, you need to multiply the corresponding terms. You can start by multiplying the terms with the same variable, and then multiply the constant terms.

Q: What if the functions have different variables?

A: If the functions have different variables, you can multiply them directly.

Q: How do I simplify the result?

A: To simplify the result, you need to combine like terms. This means combining terms with the same variable.

Q: What is the final answer?

A: The final answer is the product of the two functions, which is the resulting function after multiplying the original functions.

Q: Can I use function addition, subtraction, and multiplication to find the quotient of two functions?

A: No, function addition, subtraction, and multiplication are used to find the sum, difference, and product of two functions, not the quotient. To find the quotient of two functions, you need to use function division.

Q: How do I divide two functions?

A: To divide two functions, you need to divide the corresponding terms. You can start by dividing the terms with the same variable, and then divide the constant terms.

Q: What if the functions have different variables?

A: If the functions have different variables, you can divide them directly.

Q: How do I simplify the result?

A: To simplify the result, you need to combine like terms. This means combining terms with the same variable.

Q: What is the final answer?

A: The final answer is the quotient of the two functions, which is the resulting function after dividing the original functions.

Conclusion

In this article, we have answered some common questions about finding the sum of two functions. We have also discussed how to add, subtract, multiply, and divide functions. We hope that this article has been helpful in understanding these concepts.