In Exercises 31–50, find ƒ+g and 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 need to find the sum of two functions, \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), which is denoted as \( (f+g)(x) \).
2. Step 2: Add the functions. To find \( (f+g)(x) \), add \( f(x) \) and \( g(x) \) together: \( (f+g)(x) = f(x) + g(x) = (2x^2 - x - 3) + (x + 1) \).
3. Step 3: Combine like terms. Simplify the expression by combining like terms: \( 2x^2 - x + x - 3 + 1 \).
4. Step 4: Simplify the expression. Combine the terms to get \( 2x^2 - 2 \).
5. Step 5: Determine the domain. Since both \( f(x) \) and \( g(x) \) are polynomials, their domain is all real numbers. Therefore, the domain of \( (f+g)(x) \) is also all real numbers.

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, unless specified otherwise by 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.