Find ƒ+g, ƒ- g, ƒg and ƒ/g. Determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the sum of two functions, denoted as ƒ+g, where ƒ(x) = 2x² − x − 3 and g(x) = x + 1. Additionally, we need to determine the domain of the resulting function.
2. Step 2: Add the two functions together. To find ƒ+g, combine ƒ(x) and g(x) by adding their expressions: ƒ+g = ƒ(x) + g(x). This means we add the terms of ƒ(x) = 2x² − x − 3 and g(x) = x + 1.
3. Step 3: Combine like terms. Add the terms from ƒ(x) and g(x): (2x²) + (-x + x) + (-3 + 1). Simplify the expression to get the resulting function for ƒ+g.
4. Step 4: Determine the domain of the resulting function. The domain of a function is the set of all x-values for which the function is defined. Since both ƒ(x) and g(x) are polynomials, they are defined for all real numbers. Therefore, the domain of ƒ+g is also all real numbers.
5. Step 5: Write the final expression for ƒ+g and state the domain. The resulting function from Step 3 represents ƒ+g, and the domain is all real numbers, which can be expressed as (-∞, ∞).

Key Concepts

Function Addition

Function addition involves combining two functions by adding their outputs for each input. For functions f(x) and g(x), the sum is defined as (f + g)(x) = f(x) + g(x). This operation requires evaluating both functions at the same x-value and summing the results, which is essential for solving the given problem.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like f(x) = 2x² − x − 3 and g(x) = x + 1, the domain is typically all real numbers, as polynomials do not have restrictions such as division by zero or square roots of negative numbers.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The functions f(x) and g(x) in this problem are both polynomials, which means they are continuous and defined for all real numbers, making their analysis straightforward when determining their sum and domain.