Find fg 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. The goal is to find the composition of the functions f and g, denoted as fg(x) or f(g(x)). This means substituting g(x) into f(x).

Step 2: Write the expression for f(g(x)). Start with the definition of f(x), which is f(x) = 2x² − x − 3. Replace every occurrence of x in f(x) with g(x), which is x + 1. This gives f(g(x)) = 2(x + 1)² − (x + 1) − 3.

Step 3: Simplify the expression for f(g(x)). Begin by expanding (x + 1)² to get x² + 2x + 1. Substitute this back into the equation: f(g(x)) = 2(x² + 2x + 1) − (x + 1) − 3.

Step 4: Distribute and combine like terms. Expand 2(x² + 2x + 1) to get 2x² + 4x + 2. Then subtract (x + 1) and subtract 3. Combine all terms to simplify the expression for f(g(x)).

Step 5: Determine the domain of f(g(x)). The domain of g(x) is all real numbers since g(x) = x + 1 is a linear function. Similarly, the domain of f(x) is all real numbers since it is a quadratic function. Therefore, the domain of f(g(x)) is also all real numbers.

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주요 개념

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Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, fg means f(g(x)), which requires substituting g(x) into f(x). Understanding how to perform this substitution is crucial for finding the composed 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 composing functions, the domain of the resulting function fg must consider the domains of both f and g, ensuring that all inputs lead to valid outputs in the context of both 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 composed function fg and its domain.