What Is $ (f+g)(x) ? G I V E N : ?Given: ? G I V E N : { F(x) = -x^2 + 8x \} ${ G(x) = X + 1 }$Write Your Answer As A Polynomial Or A Rational Function In Simplest Form.

Introduction

In mathematics, the concept of function addition is a fundamental operation that allows us to combine two or more functions to create a new function. Given two functions, $f(x)$ and $g(x)$, the sum of these functions, denoted as $(f+g)(x)$, is a new function that results from adding the corresponding values of $f(x)$ and $g(x)$ for each input value of $x$. In this article, we will explore the concept of function addition and determine the value of $(f+g)(x)$ for the given functions $f(x) = -x^2 + 8x$ and $g(x) = x + 1$.

Function Addition

Function addition is a binary operation that takes two functions, $f(x)$ and $g(x)$, and produces a new function, $(f+g)(x)$, as the result. The process of function addition involves adding the corresponding values of $f(x)$ and $g(x)$ for each input value of $x$. This can be represented mathematically as:

${(f+g)(x) = f(x) + g(x)}$

Given Functions

We are given two functions, $f(x) = -x^2 + 8x$ and $g(x) = x + 1$. To find the value of $(f+g)(x)$, we need to add these two functions together.

Adding the Functions

To add the functions $f(x)$ and $g(x)$, we need to add the corresponding terms of each function. The function $f(x)$ has two terms, $-x^2$ and $8x$, while the function $g(x)$ has only one term, $x + 1$. We can add these terms together as follows:

\begin{align*} (f+g)(x) &= f(x) + g(x) \ &= (-x^2 + 8x) + (x + 1) \ &= -x^2 + 8x + x + 1 \end{align*}

Simplifying the Result

To simplify the result, we can combine like terms. The terms $8x$ and $x$ are like terms, as they both have the variable $x$ with a coefficient. We can combine these terms by adding their coefficients:

\begin{align*} (f+g)(x) &= -x^2 + 8x + x + 1 \ &= -x^2 + (8x + x) + 1 \ &= -x^2 + 9x + 1 \end{align*}

Conclusion

In conclusion, the value of $(f+g)(x)$ for the given functions $f(x) = -x^2 + 8x$ and $g(x) = x + 1$ is $-x^2 + 9x + 1$. This result is a polynomial function in simplest form.

Example Use Case

The concept of function addition has many practical applications in mathematics and other fields. For example, in physics, the sum of two functions can represent the total energy of a system. In economics, the sum of two functions can represent the total cost of a product.

Final Answer

The final answer is $\boxed{-x^2 + 9x + 1}$.

Introduction

In our previous article, we explored the concept of function addition and determined the value of $(f+g)(x)$ for the given functions $f(x) = -x^2 + 8x$ and $g(x) = x + 1$. In this article, we will answer some frequently asked questions about function addition and provide additional examples to help solidify your understanding of this concept.

Q&A

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

A: Function addition and function multiplication are two different operations that involve combining functions. Function addition involves adding the corresponding values of two functions, while function multiplication involves multiplying the corresponding values of two functions.

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

A: To add two functions with different variables, you need to use the concept of function composition. Function composition involves combining two functions by substituting one function into the other. For example, if you have two functions $f(x) = x^2$ and $g(y) = y + 1$, you can add them together by substituting $g(y)$ into $f(x)$ as follows:

${(f \circ g)(x) = f(g(x)) = (x + 1)^2}$

Q: Can I add a function to a constant?

A: Yes, you can add a function to a constant. When you add a function to a constant, the constant is treated as a function that takes no input and always returns the same value. For example, if you have a function $f(x) = x^2$ and a constant $c = 3$, you can add them together as follows:

${f(x) + c = x^2 + 3}$

Q: How do I add two functions with the same variable but different coefficients?

A: To add two functions with the same variable but different coefficients, you can simply add the coefficients of the corresponding terms. For example, if you have two functions $f(x) = 2x^2$ and $g(x) = 3x^2$, you can add them together as follows:

${f(x) + g(x) = 2x^2 + 3x^2 = 5x^2}$

Q: Can I add a function to a function with a different variable?

A: No, you cannot add a function to a function with a different variable. Function addition only works for functions with the same variable. If you have two functions with different variables, you need to use function composition to combine them.

Example Use Cases

Example 1: Adding two functions with the same variable

Suppose we have two functions $f(x) = x^2$ and $g(x) = 2x^2$. We can add them together as follows:

${f(x) + g(x) = x^2 + 2x^2 = 3x^2}$

Example 2: Adding a function to a constant

Suppose we have a function $f(x) = x^2$ and a constant $c = 3$. We can add them together as follows:

${f(x) + c = x^2 + 3}$

Example 3: Adding two functions with different variables

Suppose we have two functions $f(x) = x^2$ and $g(y) = y + 1$. We can add them together by using function composition as follows:

${(f \circ g)(x) = f(g(x)) = (x + 1)^2}$

Conclusion

In conclusion, function addition is a powerful tool that allows us to combine functions in a variety of ways. By understanding the rules of function addition, we can solve a wide range of problems in mathematics and other fields.

Final Answer

The final answer is $\boxed{3x^2}$ for the first example, $\boxed{x^2 + 3}$ for the second example, and $\boxed{(x + 1)^2}$ for the third example.