If F ( X ) = 5 X − 2 F(x)=5x-2 F ( X ) = 5 X − 2 And G ( X ) = 2 X + 1 G(x)=2x+1 G ( X ) = 2 X + 1 , Find ( F + G ) ( X (f+g)(x ( F + G ) ( X ].A. 7 X − 3 7x-3 7 X − 3 B. 3 X − 1 3x-1 3 X − 1 C. 7 X − 1 7x-1 7 X − 1 D. 4 X − 3 4x-3 4 X − 3

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Understanding the Problem

To find $(f+g)(x)$, we need to understand what the notation means. The notation $(f+g)(x)$ represents the sum of the functions $f(x)$ and $g(x)$, evaluated at the point $x$. In other words, we need to add the two functions together, term by term.

Adding the Functions

To add the functions $f(x)$ and $g(x)$, we need to add their corresponding terms. The function $f(x)$ is given by $5x-2$, and the function $g(x)$ is given by $2x+1$. We can add these two functions together by combining like terms.

Combining Like Terms

When we add the two functions together, we get:

$(f+g)(x) = (5x-2) + (2x+1)$

To combine like terms, we need to add the coefficients of the $x$ terms and the constant terms separately. The coefficient of the $x$ term in the first function is $5$, and the coefficient of the $x$ term in the second function is $2$. We can add these two coefficients together to get $7$. The constant term in the first function is $-2$, and the constant term in the second function is $1$. We can add these two constant terms together to get $-1$.

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression:

$(f+g)(x) = 7x - 1$

Conclusion

Therefore, the sum of the functions $f(x)$ and $g(x)$, evaluated at the point $x$, is given by $(f+g)(x) = 7x - 1$.

Answer

The correct answer is:

• C. $7x-1$

Explanation

The correct answer is C. $7x-1$ because this is the result of adding the two functions together, term by term, and combining like terms.

Example

To illustrate this concept, let's consider an example. Suppose we want to find $(f+g)(2)$. We can plug in $x=2$ into the expression $(f+g)(x) = 7x - 1$ to get:

$(f+g)(2) = 7(2) - 1 = 14 - 1 = 13$

This shows that the sum of the functions $f(x)$ and $g(x)$, evaluated at the point $x=2$, is equal to $13$.

Properties of Function Addition

The addition of functions has several important properties. One of the most important properties is the commutative property, which states that the order in which we add functions does not matter. In other words, $(f+g)(x) = (g+f)(x)$.

Another important property of function addition is the associative property, which states that the order in which we add functions does not matter. In other words, $(f+g+h)(x) = (f+(g+h))(x)$.

Real-World Applications

The concept of function addition has many real-world applications. For example, in economics, the sum of two functions can represent the total cost of producing a certain good or service. In physics, the sum of two functions can represent the total energy of a system.

Conclusion

In conclusion, the sum of two functions can be found by adding their corresponding terms and combining like terms. The result is a new function that represents the sum of the original functions. The concept of function addition has many real-world applications and is an important tool in mathematics and science.

Final Answer

The final answer is:

• C. $7x-1$
  

Understanding Function Addition

Function addition is a fundamental concept in mathematics that involves combining two or more functions to create a new function. In this article, we will explore the concept of function addition and answer some common questions related to it.

Q: What is function addition?

A: Function addition is the process of combining two or more functions to create a new function. It involves adding the corresponding terms of the two functions and combining like terms.

Q: How do I add two functions together?

A: To add two functions together, you need to add their corresponding terms and combine like terms. For example, if you have two functions f(x) = 3x + 2 and g(x) = 2x - 1, you can add them together by combining like terms:

(f + g)(x) = (3x + 2) + (2x - 1) = 5x + 1

Q: What is the commutative property of function addition?

A: The commutative property of function addition states that the order in which you add functions does not matter. In other words, (f + g)(x) = (g + f)(x).

Q: What is the associative property of function addition?

A: The associative property of function addition states that the order in which you add functions does not matter. In other words, (f + g + h)(x) = (f + (g + h))(x).

Q: How do I evaluate a function at a specific point?

A: To evaluate a function at a specific point, you need to plug in the value of the point into the function. For example, if you have a function f(x) = 2x + 1 and you want to evaluate it at x = 3, you can plug in x = 3 into the function:

f(3) = 2(3) + 1 = 6 + 1 = 7

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

A: Function addition and function composition are two different concepts in mathematics. Function addition involves combining two or more functions to create a new function, while function composition involves combining two or more functions to create a new function by plugging in one function into another.

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

A: Function addition has many real-world applications, including economics, physics, and engineering. For example, in economics, the sum of two functions can represent the total cost of producing a certain good or service. In physics, the sum of two functions can represent the total energy of a system.

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 evaluating the function at a specific point
• Not using the correct order of operations
• Not checking for errors in the function

Conclusion

In conclusion, function addition is a fundamental concept in mathematics that involves combining two or more functions to create a new function. By understanding the concept of function addition and answering some common questions related to it, you can better appreciate the importance of function addition in mathematics and its real-world applications.

Final Answer

The final answer is:

• C. $7x-1$

Additional Resources

For more information on function addition, please refer to the following resources:

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

Practice Problems

To practice function addition, please try the following problems:

• Add the functions f(x) = 2x + 1 and g(x) = 3x - 2.
• Evaluate the function f(x) = 2x + 1 at x = 4.
• Find the sum of the functions f(x) = 2x + 1 and g(x) = 3x - 2, evaluated at x = 4.

Answer Key

• The sum of the functions f(x) = 2x + 1 and g(x) = 3x - 2 is f(x) + g(x) = 5x - 1.
• The value of the function f(x) = 2x + 1 at x = 4 is f(4) = 2(4) + 1 = 9.
• The sum of the functions f(x) = 2x + 1 and g(x) = 3x - 2, evaluated at x = 4, is f(4) + g(4) = 9 + 5 = 14.