Given The Functions:$\[ \begin{array}{l} f(x) = 3x^3 - 2x^2 + 4x - 5 \\ g(x) = 6x - 7 \end{array} \\]Find \[$(f+g)(x)\$\].A. \[$(f+g)(x) = 3x^3 - 2x^2 + 10x - 12\$\]B. \[$(f+g)(x) = 3x^3 - 2x^2 - 2x + 12\$\]C.

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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. In this article, we will explore how to find the sum of two functions, given their individual definitions.

Understanding the Problem

We are given two functions:

f(x)=3x3−2x2+4x−5{ f(x) = 3x^3 - 2x^2 + 4x - 5 } g(x)=6x−7{ g(x) = 6x - 7 }

Our goal is to find the sum of these two functions, denoted as (f+g)(x){(f+g)(x)}.

Step 1: Understanding Function Addition

When adding two functions, we simply add their corresponding terms. This means that we will add the coefficients of the same degree of the variables.

Step 2: Adding the Functions

To find the sum of the two functions, we will add their corresponding terms:

(f+g)(x)=f(x)+g(x){ (f+g)(x) = f(x) + g(x) } (f+g)(x)=(3x3−2x2+4x−5)+(6x−7){ (f+g)(x) = (3x^3 - 2x^2 + 4x - 5) + (6x - 7) }

Step 3: Combining Like Terms

Now, we will combine the like terms:

(f+g)(x)=3x3−2x2+4x−5+6x−7{ (f+g)(x) = 3x^3 - 2x^2 + 4x - 5 + 6x - 7 } (f+g)(x)=3x3−2x2+(4x+6x)−5−7{ (f+g)(x) = 3x^3 - 2x^2 + (4x + 6x) - 5 - 7 } (f+g)(x)=3x3−2x2+10x−12{ (f+g)(x) = 3x^3 - 2x^2 + 10x - 12 }

Conclusion

In conclusion, the sum of the two functions is:

(f+g)(x)=3x3−2x2+10x−12{ (f+g)(x) = 3x^3 - 2x^2 + 10x - 12 }

This is the correct answer, which matches option A.

Discussion

The concept of adding functions is an essential part of algebra and calculus. It allows us to combine different functions to create new ones, which can be used to model real-world phenomena.

In this article, we have seen how to add two functions by combining their corresponding terms. This process is straightforward and can be applied to any two functions.

Example Use Cases

The concept of adding functions has many practical applications in various fields, including:

  • Physics: When modeling the motion of an object, we may need to combine different functions to describe its position, velocity, and acceleration.
  • Engineering: In designing electrical circuits, we may need to combine different functions to describe the behavior of the circuit.
  • Economics: When modeling economic systems, we may need to combine different functions to describe the behavior of the system.

Conclusion

In conclusion, the sum of two functions is a fundamental concept in mathematics that has many practical applications. By understanding how to add functions, we can create new functions that can be used to model real-world phenomena.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to find the sum of two functions. In this article, we will answer some frequently asked questions about finding the sum of two functions.

Q: What is the sum of two functions?

A: The sum of two functions is a new function that is created by adding the corresponding terms of the two original functions.

Q: How do I add two functions?

A: To add two functions, you simply add their corresponding terms. This means that you will add the coefficients of the same degree of the variables.

Q: What if the two functions have different variables?

A: If the two functions have different variables, you will need to substitute the variables in one function with the variables in the other function. Then, you can add the two functions.

Q: Can I add more than two functions?

A: Yes, you can add more than two functions. The process is the same as adding two functions: you simply add the corresponding terms of the functions.

Q: What if I have a function with a constant term?

A: If you have a function with a constant term, you can add it to another function by simply adding the constant term to the other function.

Q: Can I subtract one function from another?

A: Yes, you can subtract one function from another by adding the negative of the second function to the first function.

Q: What if I have a function with a variable in the denominator?

A: If you have a function with a variable in the denominator, you will need to use algebraic manipulation to simplify the function before adding it to another function.

Q: Can I add a function to a constant?

A: Yes, you can add a function to a constant by simply adding the function to the constant.

Q: What if I have a function with a trigonometric term?

A: If you have a function with a trigonometric term, you will need to use trigonometric identities to simplify the function before adding it to another function.

Q: Can I add a function to a polynomial?

A: Yes, you can add a function to a polynomial by simply adding the function to the polynomial.

Conclusion

In conclusion, finding the sum of two functions is a fundamental concept in mathematics that has many practical applications. By understanding how to add functions, we can create new functions that can be used to model real-world phenomena.

Final Answer

The final answer is:

  • Q: What is the sum of two functions? A: The sum of two functions is a new function that is created by adding the corresponding terms of the two original functions.
  • Q: How do I add two functions? A: To add two functions, you simply add their corresponding terms.
  • Q: Can I add more than two functions? A: Yes, you can add more than two functions.
  • Q: What if I have a function with a constant term? A: If you have a function with a constant term, you can add it to another function by simply adding the constant term to the other function.
  • Q: Can I subtract one function from another? A: Yes, you can subtract one function from another by adding the negative of the second function to the first function.
  • Q: What if I have a function with a variable in the denominator? A: If you have a function with a variable in the denominator, you will need to use algebraic manipulation to simplify the function before adding it to another function.
  • Q: Can I add a function to a constant? A: Yes, you can add a function to a constant by simply adding the function to the constant.
  • Q: What if I have a function with a trigonometric term? A: If you have a function with a trigonometric term, you will need to use trigonometric identities to simplify the function before adding it to another function.
  • Q: Can I add a function to a polynomial? A: Yes, you can add a function to a polynomial by simply adding the function to the polynomial.