Let ƒ(x) = 3x^2 - 4 and g(x) = x^2 - 3x -4. Find each of the following. (f+g)(2k)

Verified step by step guidance

1. Step 1: Understand that (f+g)(x) represents the sum of the functions f(x) and g(x).
2. Step 2: Write the expression for (f+g)(x) by adding the functions: (f+g)(x) = f(x) + g(x).
3. Step 3: Substitute the given functions: (f+g)(x) = (3x^2 - 4) + (x^2 - 3x - 4).
4. Step 4: Combine like terms in the expression: (f+g)(x) = 3x^2 + x^2 - 3x - 4 - 4.
5. Step 5: Substitute x with 2k in the simplified expression to find (f+g)(2k).

Key Concepts

Function Addition

Function addition involves combining two functions by adding their outputs for the same input. For functions f(x) and g(x), the sum (f+g)(x) is defined as f(x) + g(x). This concept is essential for solving the given problem, as it requires evaluating the sum of the two functions at a specific input.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this case, both f(x) and g(x) are polynomial functions, which means their behavior can be analyzed using algebraic techniques. Understanding how to manipulate and evaluate polynomials is crucial for finding (f+g)(2k).

Substitution

Substitution is the process of replacing a variable in an expression with a specific value or another expression. In this problem, we need to substitute 2k into the combined function (f+g)(x). Mastering substitution is vital for correctly evaluating the function at the given input and obtaining the final result.