Find f−g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

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Step 1: Understand the problem. You are tasked with finding the difference of two functions, f(x) and g(x), denoted as (f−g)(x). This means subtracting g(x) from f(x). Additionally, you need to determine the domain of the resulting function.

Step 2: Write the expression for (f−g)(x). Subtract g(x) from f(x): (f−g)(x) = f(x) − g(x). Substitute the given functions: f(x) = 2x² − x − 3 and g(x) = x + 1. The subtraction becomes: (f−g)(x) = (2x² − x − 3) − (x + 1).

Step 3: Simplify the expression. Distribute the negative sign across g(x): (f−g)(x) = 2x² − x − 3 − x − 1. Combine like terms: (f−g)(x) = 2x² − 2x − 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 (f−g)(x) = 2x² − 2x − 4 is a polynomial, and polynomials are defined for all real numbers, the domain is all real numbers, or (-∞, ∞).

Step 5: Summarize the results. The difference of the functions is (f−g)(x) = 2x² − 2x − 4, and the domain of the resulting function is all real numbers, (-∞, ∞).

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주요 개념

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Function Subtraction

Function subtraction involves taking two functions, f(x) and g(x), and creating a new function, f-g, defined as (f-g)(x) = f(x) - g(x). This operation requires substituting the expressions of f and g into the equation and simplifying the result to find the new 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. When subtracting functions, the domain of the resulting function f-g is determined by the intersection of the domains of f and g, ensuring that all x-values are valid for both original functions.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The function f(x) = 2x² - x - 3 is a quadratic function, and its properties, such as its vertex and axis of symmetry, can influence the overall behavior of the function when combined with another function like g(x).