The domain of a polynomial function

Oct 13, 2025 by miodeya.com 133 views

What is the Rule for ${$ (f+g)(x)$}$ and What is the Domain?

In mathematics, the concept of function addition is a fundamental operation that allows us to combine two functions into a single function. Given two functions, ${$ f(x)$}$ and ${$ g(x)$}$, the sum of these functions is defined as ${$ (f+g)(x) = f(x) + g(x)$}$. In this article, we will explore the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = 2x^2 - 8$}$ and ${$ g(x) = x + 2$}$, and determine the domain of the resulting function.

To find the rule for ${$ (f+g)(x)$}$, we need to add the two functions ${$ f(x)$}$ and ${$ g(x)$}$. This involves combining like terms and simplifying the resulting expression.

Step 1: Add the two functions

${$ (f+g)(x) = f(x) + g(x)$}{ $(f+g)(x) = (2x^2 - 8) + (x + 2)\$}$

Step 2: Combine like terms

${$ (f+g)(x) = 2x^2 - 8 + x + 2$}{ $(f+g)(x) = 2x^2 + x - 6\$}$

Therefore, the rule for ${$ (f+g)(x)$}$ is ${ $2x^2 + x - 6\$}$ .

To determine the domain of the function ${$ (f+g)(x)$}$, we need to consider the values of ${$ x$}$ for which the function is defined. In this case, the function is a polynomial, and polynomials are defined for all real numbers.

The domain of a polynomial function

A polynomial function is defined for all real numbers, which can be expressed in interval notation as ${$ (-\infty, \infty)$}$.

Therefore, the domain of the function ${$ (f+g)(x)$}$ is ${$ (-\infty, \infty)$}$.

In conclusion, the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = 2x^2 - 8$}$ and ${$ g(x) = x + 2$}$ is ${ $2x^2 + x - 6\$}$ . The domain of the resulting function is ${$ (-\infty, \infty)$}$.

Find the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = 3x^2 - 2$}$ and ${$ g(x) = 2x - 1$}$.
Determine the domain of the function ${$ (f+g)(x)$}$ when ${$ f(x) = x^2 + 1$}$ and ${$ g(x) = 2x - 3$}$.

${$ (f+g)(x) = (3x^2 - 2) + (2x - 1)$}{ $(f+g)(x) = 3x^2 + 2x - 3\$}$ The domain of the function is ${$ (-\infty, \infty)$}$.
${$ (f+g)(x) = (x^2 + 1) + (2x - 3)$}{ $(f+g)(x) = x^2 + 2x - 2\$}$ The domain of the function is ${$ (-\infty, \infty)$}$.

In our previous article, we explored the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = 2x^2 - 8$}$ and ${$ g(x) = x + 2$}$, and determined the domain of the resulting function. In this article, we will answer some frequently asked questions about function addition and domain.

Q: What is the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = ax^2 + bx + c$}$ and ${$ g(x) = dx^2 + ex + f$}$?

A: To find the rule for ${$ (f+g)(x)$}$, we need to add the two functions ${$ f(x)$}$ and ${$ g(x)$}$. This involves combining like terms and simplifying the resulting expression.

${$ (f+g)(x) = (ax^2 + bx + c) + (dx^2 + ex + f)$}{ $(f+g)(x) = (a+d)x^2 + (b+e)x + (c+f)\$}$

Q: What is the domain of the function ${$ (f+g)(x)$}$ when ${$ f(x) = x^2 + 1$}$ and ${$ g(x) = 2x - 3$}$?

A: The domain of a polynomial function is all real numbers, which can be expressed in interval notation as ${$ (-\infty, \infty)$}$.

Q: Can we add two functions that have different domains?

A: Yes, we can add two functions that have different domains. However, the resulting function will have the same domain as the function with the most restrictive domain.

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

A: To determine the domain of a function, we need to consider the values of ${$ x$}$ for which the function is defined. In the case of polynomial functions, the domain is all real numbers. However, for other types of functions, such as rational functions, the domain may be restricted to certain intervals or sets of numbers.

Q: Can we add two functions that have different variables?

A: No, we cannot add two functions that have different variables. For example, we cannot add ${$ f(x) = x^2 + 1$}$ and ${$ g(y) = 2y - 3$}$ because they have different variables.

Q: How do we simplify the resulting expression when adding two functions?

A: To simplify the resulting expression, we need to combine like terms and eliminate any unnecessary parentheses or other mathematical notation.

Q: Can we add two functions that have different coefficients?

A: Yes, we can add two functions that have different coefficients. For example, we can add ${$ f(x) = 2x^2 + 3x + 1$}$ and ${$ g(x) = 4x^2 - 2x + 3$}$.

Q: How do we determine the domain of a function with a variable in the denominator?

A: To determine the domain of a function with a variable in the denominator, we need to consider the values of ${$ x$}$ for which the denominator is not equal to zero.

In conclusion, function addition and domain are important concepts in mathematics. By understanding how to add two functions and determine their domain, we can solve a wide range of problems in algebra, calculus, and other areas of mathematics.

Find the rule for ${$ (f+g)(x)$}$ when ${$ f(x) = 3x^2 - 2$}$ and ${$ g(x) = 2x - 1$}$.
Determine the domain of the function ${$ (f+g)(x)$}$ when ${$ f(x) = x^2 + 1$}$ and ${$ g(x) = 2x - 3$}$.

${$ (f+g)(x) = (3x^2 - 2) + (2x - 1)$}{ $(f+g)(x) = 3x^2 + 2x - 3\$}$ The domain of the function is ${$ (-\infty, \infty)$}$.
${$ (f+g)(x) = (x^2 + 1) + (2x - 3)$}{ $(f+g)(x) = x^2 + 2x - 2\$}$ The domain of the function is ${$ (-\infty, \infty)$}$.

In this article, we answered some frequently asked questions about function addition and domain. We also provided example problems and solutions to reinforce the concepts. By understanding how to add two functions and determine their domain, we can solve a wide range of problems in mathematics.