What Is { (f+g)(x)$} ? G I V E N : ?Given: ? G I V E N : { \begin{align*} f(x) &= X^3 - X \\ g(x) &= X^3 + 2x^2 - 10 \end{align*} \} Enter Your Answer In Standard Form In The Box. { (f+g)(x) =$}$ { \square$}$

by ADMIN 211 views

Introduction to Function Addition

In mathematics, functions are a fundamental concept that helps us describe relationships between variables. When we add two functions, we are essentially combining their outputs to create a new function. This process is called function addition, and it's a crucial operation in algebra and calculus. In this article, we will explore the concept of function addition and learn how to add two functions, f(x)f(x) and g(x)g(x).

Understanding the Given Functions

We are given two functions:

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

These functions are defined for all real values of xx. Our task is to find the sum of these two functions, denoted as (f+g)(x)(f+g)(x).

Adding Functions

To add two functions, we simply add their corresponding terms. In this case, we have:

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

Simplifying the Expression

Now, let's simplify the expression by combining like terms:

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

Standard Form

The standard form of a polynomial function is a sum of terms, where each term is a product of a coefficient and a variable raised to a non-negative integer power. In our case, the standard form of (f+g)(x)(f+g)(x) is:

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

Conclusion

In this article, we learned how to add two functions, f(x)f(x) and g(x)g(x). We saw that the sum of these two functions is a new function, denoted as (f+g)(x)(f+g)(x). We also simplified the expression and wrote it in standard form. This process is essential in algebra and calculus, as it helps us combine functions and create new ones.

Examples and Applications

Function addition has numerous applications in mathematics and science. Here are a few examples:

  • Physics: When two physical systems interact, their energies can be combined to form a new energy function.
  • Economics: In economics, the demand and supply functions can be combined to form a new function that represents the market equilibrium.
  • Computer Science: In computer science, function addition is used in algorithms and data structures to combine functions and create new ones.

Tips and Tricks

When adding functions, remember to:

  • Combine like terms: Combine terms with the same variable and exponent.
  • Simplify the expression: Simplify the expression by combining like terms and removing any unnecessary terms.
  • Write in standard form: Write the final expression in standard form, which is a sum of terms, where each term is a product of a coefficient and a variable raised to a non-negative integer power.

Final Answer

The final answer is:

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

Introduction

In our previous article, we explored the concept of function addition and learned how to add two functions, f(x)f(x) and g(x)g(x). In this article, we will answer some frequently asked questions about function addition and provide additional examples and explanations.

Q1: What is the difference between function addition and function multiplication?

A1: Function addition and function multiplication are two different operations. Function addition involves combining the outputs of two functions to create a new function, while function multiplication involves combining the outputs of two functions to create a new function that is scaled by a constant factor.

Q2: How do I add two functions with different variables?

A2: When adding two functions with different variables, you need to use the concept of function composition. Function composition involves combining two functions to create a new function that takes the output of one function as the input of the other function.

Q3: Can I add two functions that are not defined for the same values of x?

A3: No, you cannot add two functions that are not defined for the same values of x. When adding two functions, you need to ensure that both functions are defined for the same values of x.

Q4: How do I simplify the expression of a function that is the sum of two other functions?

A4: To simplify the expression of a function that is the sum of two other functions, you need to combine like terms and remove any unnecessary terms.

Q5: Can I add two functions that have different domains?

A5: No, you cannot add two functions that have different domains. When adding two functions, you need to ensure that both functions have the same domain.

Q6: How do I determine the range of a function that is the sum of two other functions?

A6: To determine the range of a function that is the sum of two other functions, you need to consider the ranges of the individual functions and the properties of the sum.

Q7: Can I add two functions that are not continuous?

A7: No, you cannot add two functions that are not continuous. When adding two functions, you need to ensure that both functions are continuous.

Q8: How do I graph a function that is the sum of two other functions?

A8: To graph a function that is the sum of two other functions, you need to graph the individual functions and combine them to create a new graph.

Q9: Can I add two functions that have different degrees?

A9: Yes, you can add two functions that have different degrees. However, you need to ensure that the resulting function has a well-defined degree.

Q10: How do I determine the degree of a function that is the sum of two other functions?

A10: To determine the degree of a function that is the sum of two other functions, you need to consider the degrees of the individual functions and the properties of the sum.

Conclusion

In this article, we answered some frequently asked questions about function addition and provided additional examples and explanations. We hope that this article has helped you to better understand the concept of function addition and how to apply it in different situations.

Examples and Applications

Function addition has numerous applications in mathematics and science. Here are a few examples:

  • Physics: When two physical systems interact, their energies can be combined to form a new energy function.
  • Economics: In economics, the demand and supply functions can be combined to form a new function that represents the market equilibrium.
  • Computer Science: In computer science, function addition is used in algorithms and data structures to combine functions and create new ones.

Tips and Tricks

When adding functions, remember to:

  • Combine like terms: Combine terms with the same variable and exponent.
  • Simplify the expression: Simplify the expression by combining like terms and removing any unnecessary terms.
  • Write in standard form: Write the final expression in standard form, which is a sum of terms, where each term is a product of a coefficient and a variable raised to a non-negative integer power.

Final Answer

The final answer is:

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