Consider The Following Functions:$ \begin{array}{c} f = {(-4,-2), (2,-3), (3,0)} \ g = {(2,1), (-2,-4)} \end{array} }$Step 1 Of 4 Find { (f+g)(2)$ $.

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In mathematics, functions are used to describe the relationship between two variables. When dealing with functions, it's essential to understand how to perform various operations, including addition. In this article, we will explore the concept of function addition and how to apply it to find the value of a function at a specific point.

What is Function Addition?

Function addition is a mathematical operation that combines two or more functions to create a new function. This operation is denoted by the symbol "+" and is used to add the values of the two functions at a specific point. The resulting function is a new function that combines the characteristics of the original functions.

Defining the Functions

In this problem, we are given two functions, f and g, defined as follows:

  • Function f: f = {(-4,-2), (2,-3), (3,0)}
  • Function g: g = {(2,1), (-2,-4)}

These functions are defined as sets of ordered pairs, where each pair represents the input and output values of the function.

Step 1: Finding the Value of (f+g)(2)

To find the value of (f+g)(2), we need to add the values of functions f and g at the point x = 2.

Function f at x = 2: Since the point (2,-3) is in the set of ordered pairs for function f, we can conclude that f(2) = -3.

Function g at x = 2: Since the point (2,1) is in the set of ordered pairs for function g, we can conclude that g(2) = 1.

Adding the Values: Now that we have the values of functions f and g at x = 2, we can add them to find the value of (f+g)(2).

(f+g)(2) = f(2) + g(2) = -3 + 1 = -2

Therefore, the value of (f+g)(2) is -2.

Conclusion

In this article, we explored the concept of function addition and how to apply it to find the value of a function at a specific point. We defined two functions, f and g, and used them to find the value of (f+g)(2). The resulting value was -2, which demonstrates the importance of understanding function addition in mathematics.

Future Steps

In the next step, we will explore how to find the value of (f+g)(-2) using the same functions.

Step 2: Finding the Value of (f+g)(-2)

To find the value of (f+g)(-2), we need to add the values of functions f and g at the point x = -2.

Function f at x = -2: Since the point (-4,-2) is in the set of ordered pairs for function f, we can conclude that f(-2) = -2.

Function g at x = -2: Since the point (-2,-4) is in the set of ordered pairs for function g, we can conclude that g(-2) = -4.

Adding the Values: Now that we have the values of functions f and g at x = -2, we can add them to find the value of (f+g)(-2).

(f+g)(-2) = f(-2) + g(-2) = -2 + (-4) = -6

Therefore, the value of (f+g)(-2) is -6.

Step 3: Finding the Value of (f+g)(3)

To find the value of (f+g)(3), we need to add the values of functions f and g at the point x = 3.

Function f at x = 3: Since the point (3,0) is in the set of ordered pairs for function f, we can conclude that f(3) = 0.

Function g at x = 3: Since the point (2,1) is in the set of ordered pairs for function g, we can conclude that g(3) is not defined.

Adding the Values: Since g(3) is not defined, we cannot add the values of functions f and g at x = 3.

Therefore, the value of (f+g)(3) is undefined.

Step 4: Conclusion

In this article, we explored the concept of function addition and how to apply it to find the value of a function at a specific point. We defined two functions, f and g, and used them to find the values of (f+g)(2), (f+g)(-2), and (f+g)(3). The resulting values were -2, -6, and undefined, respectively, which demonstrates the importance of understanding function addition in mathematics.

Future Directions

In this article, we will answer some frequently asked questions about function addition.

Q: What is function addition?

A: Function addition is a mathematical operation that combines two or more functions to create a new function. This operation is denoted by the symbol "+" and is used to add the values of the two functions at a specific point.

Q: How do I add two functions?

A: To add two functions, you need to add the values of the two functions at each point. For example, if you have two functions f and g, and you want to find the value of (f+g)(x), you need to add the values of f(x) and g(x) at each point x.

Q: What if one of the functions is not defined at a point?

A: If one of the functions is not defined at a point, then the value of the sum of the two functions at that point is also not defined. For example, if you have two functions f and g, and f(x) is not defined, then (f+g)(x) is also not defined.

Q: Can I add more than two functions?

A: Yes, you can add more than two functions. For example, if you have three functions f, g, and h, you can find the value of (f+g+h)(x) by adding the values of f(x), g(x), and h(x) at each point x.

Q: How do I use function addition in real-world problems?

A: Function addition can be used to solve a wide range of real-world problems, such as modeling population growth, predicting stock prices, and analyzing data. For example, if you are modeling the growth of a population, you can use function addition to combine the growth rates of different populations.

Q: What are some common applications of function addition?

A: Some common applications of function addition include:

  • Modeling population growth
  • Predicting stock prices
  • Analyzing data
  • Solving optimization problems
  • Modeling complex systems

Q: What are some common mistakes to avoid when using function addition?

A: Some common mistakes to avoid when using function addition include:

  • Not defining the functions correctly
  • Not checking for undefined values
  • Not using the correct notation
  • Not considering the domain and range of the functions

Q: How do I check my work when using function addition?

A: To check your work when using function addition, you can:

  • Verify that the functions are defined correctly
  • Check for undefined values
  • Use a calculator or computer program to check the results
  • Compare your results with the expected results

Conclusion

In this article, we answered some frequently asked questions about function addition. We discussed how to add two or more functions, how to handle undefined values, and how to use function addition in real-world problems. We also provided some common applications and mistakes to avoid when using function addition. By following these guidelines, you can use function addition to solve a wide range of problems in mathematics and real-world applications.

Future Directions

In the future, we can explore more advanced topics in function addition, such as finding the sum of multiple functions, using function addition to solve optimization problems, and applying function addition to real-world problems.

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