For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2.
ƒ(x)=2x^2-3x, g(x)=x^2-x+3

검증된 단계별 안내

Identify the given functions: \( f(x) = 2x^2 - 3x \) and \( g(x) = x^2 - x + 3 \).

Find the composite function \( (f \circ g)(x) \), which means \( f(g(x)) \). This involves substituting \( g(x) \) into every \( x \) in \( f(x) \).

Write the expression for \( f(g(x)) \) by replacing \( x \) in \( f(x) \) with \( g(x) \): \[ f(g(x)) = 2(g(x))^2 - 3(g(x)) \].

Simplify the expression by first squaring \( g(x) = x^2 - x + 3 \), then multiply by 2, and subtract 3 times \( g(x) \).

Determine the domain of each function: since both \( f(x) \) and \( g(x) \) are polynomials, their domains are all real numbers. For the composite \( (f \circ g)(x) \), the domain is all real numbers as well.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Composition

Function composition involves applying one function to the result of another, denoted as (ƒg)(x) = ƒ(g(x)). This means you first evaluate g(x), then substitute that output into ƒ. Understanding this process is essential to correctly find the composite function.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of (ƒg)(x) depends on the domain of g and the domain of ƒ evaluated at g(x). Identifying these domains ensures the composite function is valid.

Polynomial Functions

Both ƒ(x) and g(x) are polynomial functions, which are defined for all real numbers. Recognizing this helps simplify domain considerations, as polynomials have no restrictions like division by zero or square roots of negative numbers.

For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3