In Exercises 31–50, find ƒ+g and determine the domain for each function. f(x) = 2x + 3, g(x) = x − 1

Verified step by step guidance

1. Step 1: Understand that the problem requires finding the sum of two functions, \( f(x) \) and \( g(x) \).
2. Step 2: Write the expressions for the functions: \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \).
3. Step 3: To find \( (f+g)(x) \), add the two functions together: \( (f+g)(x) = f(x) + g(x) = (2x + 3) + (x - 1) \).
4. Step 4: Simplify the expression by combining like terms: \( (f+g)(x) = 2x + x + 3 - 1 \).
5. Step 5: Determine the domain of \( f+g \). Since both \( f(x) \) and \( g(x) \) are linear functions, their domain is all real numbers, \( \mathbb{R} \).

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). In this case, you would calculate f(x) = 2x + 3 and g(x) = x - 1, then add these expressions together to find the resulting function.

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) and g(x), the domain is typically all real numbers. However, when combining functions, it is essential to ensure that the resulting function's domain is consistent with the domains of the individual functions.

Linear Functions

Linear functions are mathematical expressions that create a straight line when graphed. They can be represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this problem, both f(x) and g(x) are linear functions, which simplifies the process of finding their sum and determining the domain.