Find ƒ+g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the sum of two functions, ƒ(x) and g(x), denoted as (ƒ+g)(x). This means you need to add the expressions for ƒ(x) and g(x) together.
2. Step 2: Write the sum of the functions. The sum of ƒ(x) = x - 5 and g(x) = 3x² is (ƒ+g)(x) = ƒ(x) + g(x). Substitute the given expressions: (ƒ+g)(x) = (x - 5) + (3x²).
3. Step 3: Combine like terms. Rearrange the terms in descending order of the powers of x: (ƒ+g)(x) = 3x² + x - 5.
4. Step 4: Determine the domain of the combined function. The domain of a function is the set of all x-values for which the function is defined. Both ƒ(x) and g(x) are polynomials, and polynomials are defined for all real numbers. Therefore, the domain of (ƒ+g)(x) is all real numbers, or (-∞, ∞).
5. Step 5: Summarize the result. The combined function is (ƒ+g)(x) = 3x² + x - 5, and its domain is 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) = x - 5 and g(x) = 3x², the domain is typically all real numbers, as there are no restrictions such as division by zero or square roots of negative numbers. Understanding the domain is crucial for determining valid inputs for the combined function.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this case, f(x) = x - 5 is a linear polynomial, and g(x) = 3x² is a quadratic polynomial. Recognizing the types of polynomials helps in understanding their behavior, including their domains and how they can be combined.