Calculate { (f+g)(x)$}$ For { F(x) = 2x^2 - 8$}$ And { G(x) = X + 2$}$.Choose The Correct Expression:A. ${ 2x^2 - X - 6\$} B. ${ 4x^2\$} C. ${ 3x^2 - 6\$} D. ${ 2x^2 + X - 6\$}
Introduction
In mathematics, functions are a fundamental concept that helps us describe and analyze relationships between variables. When we have two functions, we can combine them to create a new function, known as the sum of the two functions. In this article, we will explore how to calculate the sum of two functions, using the given functions {f(x) = 2x^2 - 8$}$ and {g(x) = x + 2$}$ as examples.
Understanding Function Summation
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. In other words, we add the x-terms and the constant terms separately. This concept is based on the principle of function addition, which states that the sum of two functions is a new function that is equal to the sum of the individual functions.
Calculating the Sum of the Given Functions
To calculate the sum of the given functions, we need to add the corresponding terms of the two functions. Let's start by adding the x-terms:
{(f+g)(x) = (2x^2 - 8) + (x + 2)$}$
Next, we need to add the constant terms:
{(f+g)(x) = 2x^2 - 8 + x + 2$}$
Now, we can combine the like terms:
{(f+g)(x) = 2x^2 + x - 6$}$
Evaluating the Answer Choices
Now that we have calculated the sum of the given functions, let's evaluate the answer choices:
- A. ${2x^2 - x - 6\$}: This is not the correct answer, as it does not match the calculated sum.
- B. ${4x^2\$}: This is not the correct answer, as it does not match the calculated sum.
- C. ${3x^2 - 6\$}: This is not the correct answer, as it does not match the calculated sum.
- D. ${2x^2 + x - 6\$}: This is the correct answer, as it matches the calculated sum.
Conclusion
In conclusion, calculating the sum of two functions involves adding the corresponding terms of the two functions. By following the steps outlined in this article, we can calculate the sum of the given functions and evaluate the answer choices. The correct answer is ${2x^2 + x - 6\$}.
Additional Examples
To further illustrate the concept of function summation, let's consider a few additional examples:
- {(f+g)(x) = (3x^2 + 2) + (x - 1)$}$
- {(f+g)(x) = (2x^2 - 4) + (x + 3)$}$
- {(f+g)(x) = (x^2 + 1) + (2x - 2)$}$
In each of these examples, we can calculate the sum of the two functions by adding the corresponding terms.
Tips and Tricks
When calculating the sum of two functions, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate any expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following these steps, we can ensure that we calculate the sum of the two functions correctly.
Common Mistakes
When calculating the sum of two functions, it's easy to make mistakes. Here are a few common mistakes to watch out for:
- Forgetting to add the constant terms.
- Not combining like terms correctly.
- Not following the order of operations (PEMDAS).
By being aware of these common mistakes, we can avoid them and ensure that we calculate the sum of the two functions correctly.
Real-World Applications
The concept of function summation has numerous real-world applications. Here are a few examples:
- Physics: When calculating the motion of an object, we often need to sum the velocities of the object at different times.
- Engineering: When designing a system, we may need to sum the forces acting on the system to determine its overall behavior.
- Economics: When analyzing the behavior of a market, we may need to sum the demand and supply functions to determine the equilibrium price.
In each of these examples, the concept of function summation is essential for understanding the behavior of the system.
Conclusion
Introduction
In our previous article, we explored how to calculate the sum of two functions, using the given functions {f(x) = 2x^2 - 8$}$ and {g(x) = x + 2$}$ as examples. In this article, we will provide a Q&A guide to help you better understand the concept of function summation.
Q: What is the sum of two functions?
A: 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.
Q: How do I calculate the sum of two functions?
A: To calculate the sum of two functions, you need to add the corresponding terms of the two functions. This involves adding the x-terms and the constant terms separately.
Q: What is the order of operations for calculating the sum of two functions?
A: The order of operations for calculating the sum of two functions is:
- Parentheses: Evaluate any expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: What are some common mistakes to watch out for when calculating the sum of two functions?
A: Some common mistakes to watch out for when calculating the sum of two functions include:
- Forgetting to add the constant terms.
- Not combining like terms correctly.
- Not following the order of operations (PEMDAS).
Q: What are some real-world applications of function summation?
A: The concept of function summation has numerous real-world applications, including:
- Physics: When calculating the motion of an object, we often need to sum the velocities of the object at different times.
- Engineering: When designing a system, we may need to sum the forces acting on the system to determine its overall behavior.
- Economics: When analyzing the behavior of a market, we may need to sum the demand and supply functions to determine the equilibrium price.
Q: How do I evaluate the answer choices for a function summation problem?
A: To evaluate the answer choices for a function summation problem, you need to calculate the sum of the two functions and compare it to the answer choices. Make sure to follow the order of operations and combine like terms correctly.
Q: What is the correct answer for the function summation problem {(f+g)(x) = (2x^2 - 8) + (x + 2)$}$?
A: The correct answer for the function summation problem {(f+g)(x) = (2x^2 - 8) + (x + 2)$}$ is ${2x^2 + x - 6\$}.
Q: How do I use function summation in real-world applications?
A: To use function summation in real-world applications, you need to identify the functions that need to be summed and calculate the sum of the functions. This involves following the order of operations and combining like terms correctly.
Conclusion
In conclusion, calculating the sum of two functions is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can calculate the sum of two functions and evaluate the answer choices. Remember to follow the order of operations and combine like terms correctly to ensure accurate results.
Additional Resources
For more information on function summation, check out the following resources:
- Mathematics textbooks: Many mathematics textbooks cover the concept of function summation in detail.
- Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises on function summation.
- Mathematical software: Software such as Mathematica and Maple can be used to calculate the sum of two functions and visualize the results.
By following the steps outlined in this article and using the additional resources provided, you can master the concept of function summation and apply it to real-world problems.