Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=−6x+9,g(x)=5x+7ƒ(x)=-6x+9, g(x)=5x+7

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 73a

3장, 문제 73a

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = - 6 x + 9, g (x) = 5 x + 7$

검증된 단계별 안내

Identify the given functions: $f(x) = -6x + 9$ and $g(x) = 5x + 7$.

Recall that the composition $(f \circ g)(x)$ 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 write $f(g(x)) = -6(g(x)) + 9$.

Simplify the expression by distributing and combining like terms: $f(g(x)) = -6(5x + 7) + 9$.

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 combine the two given functions.

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 find valid inputs. Identifying these restrictions ensures the composite function is well-defined.

Linear Functions and Their Properties

Both f(x) = -6x + 9 and g(x) = 5x + 7 are linear functions, which are defined for all real numbers. Knowing that linear functions have no domain restrictions simplifies finding the domain of the composite function, as the domain will typically be all real numbers unless otherwise specified.