Find The Formulas For The Functions $f+g$, $f-g$, And $f \cdot G$. Give The Domain Of Each.Given:$\[ F(x) = X^2 \\]$\[ G(x) = 2x - 9 \\]1. $f+g = \square$ - (Simplify Your Answer.) - The Domain

Finding Formulas for Functions: $f+g$, $f-g$, and $f \cdot g$

In this article, we will explore the process of finding formulas for the functions $f+g$, $f-g$, and $f \cdot g$, given two functions $f(x)$ and $g(x)$. We will also determine the domain of each resulting function. The functions $f(x)$ and $g(x)$ are defined as:

${ f(x) = x^2 }$

${ g(x) = 2x - 9 }$

Finding the Formula for $f+g$

To find the formula for $f+g$, we need to add the two functions together. This means that we will add the corresponding terms of each function.

${ f(x) + g(x) = x^2 + (2x - 9) }$

Using the distributive property, we can simplify the expression by distributing the negative sign to the terms inside the parentheses.

${ f(x) + g(x) = x^2 + 2x - 9 }$

This is the formula for $f+g$.

Finding the Domain of $f+g$

To find the domain of $f+g$, we need to consider the values of $x$ for which the function is defined. Since the function $f+g$ is a polynomial, it is defined for all real numbers. Therefore, the domain of $f+g$ is the set of all real numbers, which can be represented as:

${ (-\infty, \infty) }$

Finding the Formula for $f-g$

To find the formula for $f-g$, we need to subtract the two functions. This means that we will subtract the corresponding terms of each function.

${ f(x) - g(x) = x^2 - (2x - 9) }$

Using the distributive property, we can simplify the expression by distributing the negative sign to the terms inside the parentheses.

${ f(x) - g(x) = x^2 - 2x + 9 }$

This is the formula for $f-g$.

Finding the Domain of $f-g$

To find the domain of $f-g$, we need to consider the values of $x$ for which the function is defined. Since the function $f-g$ is a polynomial, it is defined for all real numbers. Therefore, the domain of $f-g$ is the set of all real numbers, which can be represented as:

${ (-\infty, \infty) }$

Finding the Formula for $f \cdot g$

To find the formula for $f \cdot g$, we need to multiply the two functions together. This means that we will multiply the corresponding terms of each function.

${ f(x) \cdot g(x) = x^2 \cdot (2x - 9) }$

Using the distributive property, we can simplify the expression by distributing the terms inside the parentheses.

${ f(x) \cdot g(x) = 2x^3 - 9x^2 }$

This is the formula for $f \cdot g$.

Finding the Domain of $f \cdot g$

To find the domain of $f \cdot g$, we need to consider the values of $x$ for which the function is defined. Since the function $f \cdot g$ is a polynomial, it is defined for all real numbers. Therefore, the domain of $f \cdot g$ is the set of all real numbers, which can be represented as:

${ (-\infty, \infty) }$

In this article, we have found the formulas for the functions $f+g$, $f-g$, and $f \cdot g$, given the functions $f(x) = x^2$ and $g(x) = 2x - 9$. We have also determined the domain of each resulting function, which is the set of all real numbers. These results demonstrate the importance of understanding the properties of functions and how they can be combined to create new functions.

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Functions, 1st edition, Michael Spivak

• Domain: The set of all possible input values for a function.
• Function: A relation between a set of inputs (called the domain) and a set of possible outputs.
• Polynomial: A function of the form $ax^n + bx^{n-1} + \cdots + cx + d$, where $a, b, \ldots, c, d$ are constants and $n$ is a non-negative integer.
  Q&A: Finding Formulas for Functions $f+g$, $f-g$, and $f \cdot g$

In our previous article, we explored the process of finding formulas for the functions $f+g$, $f-g$, and $f \cdot g$, given two functions $f(x)$ and $g(x)$. We also determined the domain of each resulting function. In this article, we will answer some frequently asked questions related to finding formulas for functions.

Q: What is the formula for $f+g$?

A: The formula for $f+g$ is $f(x) + g(x) = x^2 + (2x - 9)$. This can be simplified to $x^2 + 2x - 9$.

Q: What is the domain of $f+g$?

A: The domain of $f+g$ is the set of all real numbers, which can be represented as $(-\infty, \infty)$.

Q: What is the formula for $f-g$?

A: The formula for $f-g$ is $f(x) - g(x) = x^2 - (2x - 9)$. This can be simplified to $x^2 - 2x + 9$.

Q: What is the domain of $f-g$?

A: The domain of $f-g$ is the set of all real numbers, which can be represented as $(-\infty, \infty)$.

Q: What is the formula for $f \cdot g$?

A: The formula for $f \cdot g$ is $f(x) \cdot g(x) = x^2 \cdot (2x - 9)$. This can be simplified to $2x^3 - 9x^2$.

Q: What is the domain of $f \cdot g$?

A: The domain of $f \cdot g$ is the set of all real numbers, which can be represented as $(-\infty, \infty)$.

Q: Can I use the same formula for $f+g$ and $f-g$?

A: No, the formulas for $f+g$ and $f-g$ are different. The formula for $f+g$ is $x^2 + 2x - 9$, while the formula for $f-g$ is $x^2 - 2x + 9$.

Q: Can I use the same formula for $f \cdot g$ and $f+g$?

A: No, the formulas for $f \cdot g$ and $f+g$ are different. The formula for $f \cdot g$ is $2x^3 - 9x^2$, while the formula for $f+g$ is $x^2 + 2x - 9$.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the values of $x$ for which the function is defined. For polynomial functions, the domain is usually the set of all real numbers.

Q: What is the difference between a polynomial and a non-polynomial function?

A: A polynomial function is a function of the form $ax^n + bx^{n-1} + \cdots + cx + d$, where $a, b, \ldots, c, d$ are constants and $n$ is a non-negative integer. A non-polynomial function is any function that is not a polynomial.

In this article, we have answered some frequently asked questions related to finding formulas for functions. We have also reviewed the formulas for $f+g$, $f-g$, and $f \cdot g$, as well as the domains of each resulting function. We hope that this article has been helpful in clarifying any confusion related to finding formulas for functions.

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Functions, 1st edition, Michael Spivak

• Domain: The set of all possible input values for a function.
• Function: A relation between a set of inputs (called the domain) and a set of possible outputs.
• Polynomial: A function of the form $ax^n + bx^{n-1} + \cdots + cx + d$, where $a, b, \ldots, c, d$ are constants and $n$ is a non-negative integer.