Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=8x+12,g(x)=3x−1ƒ(x)=8x+12, g(x)=3x-1

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 74a

3장, 문제 74a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = 8 x + 12, g (x) = 3 x - 1$

검증된 단계별 안내

Identify the given functions: $f(x) = 8x + 12$ and $g(x) = 3x - 1$.

Recall that the composition of functions $(f \circ g)(x)$ means $f(g(x))$, which is the function $f$ evaluated at $g(x)$.

Substitute $g(x)$ into $f(x)$: replace every $x$ in $f(x)$ with $g(x)$, so $(f \circ g)(x) = f(3x - 1) = 8(3x - 1) + 12$.

Simplify the expression by distributing and combining like terms to write $(f \circ g)(x)$ in standard form.

Determine the domain of $(f \circ g)(x)$ by considering the domain of $g(x)$ and the domain of $f$ evaluated at $g(x)$. Since both $f$ and $g$ are linear functions, 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(x), then use that output as the input for f. Understanding this process is essential to correctly find (f∘g)(x).

Domain of a Composite Function

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. This means you must consider restrictions from both functions to determine where the composition is defined.

Linear Functions

Both f(x) = 8x + 12 and g(x) = 3x - 1 are linear functions, which are defined for all real numbers. Recognizing this helps simplify finding the domain of the composite function, as there are no domain restrictions from either function.