Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=$$x^3$$, g(x)=$$x^2+3x-1$$

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 77a

3장, 문제 77a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = x^{3}, g (x) = x^{2} + 3 x - 1$

검증된 단계별 안내

Identify the given functions: $f(x) = x^3$ and $g(x) = x^2 + 3x - 1$.

Recall that the composition of functions $(f \circ g)(x)$ means $f(g(x))$, which is applying $g$ first, then $f$ to the result.

Substitute $g(x)$ into $f$: write $(f \circ g)(x) = f(g(x)) = (g(x))^3$.

Express $(f \circ g)(x)$ explicitly by cubing the expression for $g(x)$: $(x^2 + 3x - 1)^3$.

Determine the domain of $(f \circ g)(x)$ by considering the domain of $g(x)$ and the domain of $f$ applied to $g(x)$. Since both $f$ and $g$ are polynomials, their domains are all real numbers, so the domain of $(f \circ g)(x)$ is all real numbers.

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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 (f∘g)(x) = f(g(x)). It means you first evaluate g at x, then use that output as the input for f. Understanding this process is essential to correctly find (f∘g)(x).

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents with coefficients. Here, f(x) = x^3 and g(x) = x^2 + 3x - 1 are polynomials, which are continuous and defined for all real numbers. Recognizing their form helps in simplifying and composing the functions.

Domain of Composite Functions

The domain of (f∘g)(x) consists of all x-values in the domain of g for which g(x) is in the domain of f. Since polynomials are defined for all real numbers, the domain is typically all real numbers unless restricted by the composition. Identifying the domain ensures the composite function is valid.

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=x3,g(x)=x2+3x−1ƒ(x)=x^3, g(x)=x^2+3x-1