Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√x,g(x)=x+3ƒ(x)=√x, g(x)=x+3

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 75b

3장, 문제 75b

Given functions f and g, find (b) $(g \circ ƒ) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = √ x, g (x) = x + 3$

검증된 단계별 안내

Identify the given functions: $ f(x) = \sqrt{x} $ and $ g(x) = x + 3 $.

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

Substitute $ f(x) $ into $ g $: $ (g \circ f)(x) = g(\sqrt{x}) = \sqrt{x} + 3 $.

Determine the domain of $ (g \circ f)(x) $ by considering the domain of $ f(x) $ first, since $ f $ is applied first. Since $ f(x) = \sqrt{x} $, the domain is $ x \geq 0 $ because the square root requires non-negative inputs.

Since $ g(x) = x + 3 $ is defined for all real numbers, the domain of $ (g \circ f)(x) $ is the same as the domain of $ f(x) $, which is $ x \geq 0 $.

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

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When composing functions, the domain of (g∘f) depends on the domain of f and the domain restrictions of g applied to f(x). Identifying these restrictions ensures the composition is valid.

Square Root Function Domain

The square root function ƒ(x) = √x is defined only for x ≥ 0 because the square root of a negative number is not a real number. This domain restriction affects the composition since f(x) must produce values within g's domain, and x must satisfy x ≥ 0.