Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=x-1

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Identify the given functions: \(f(x) = \sqrt{x}\) and \(g(x) = x - 1\).

For part (a), find the composition \((f \circ g)(x)\), which means \(f(g(x))\). Substitute \(g(x)\) into \(f\): write \(f(g(x)) = f(x - 1) = \sqrt{x - 1}\).

Determine the domain of \((f \circ g)(x)\). Since \(f(x) = \sqrt{x}\) requires the input to be greater than or equal to zero, set the inside of the square root \(x - 1 \geq 0\) and solve for \(x\).

For part (b), find the composition \((g \circ f)(x)\), which means \(g(f(x))\). Substitute \(f(x)\) into \(g\): write \(g(f(x)) = g(\sqrt{x}) = \sqrt{x} - 1\).

Determine the domain of \((g \circ f)(x)\). Since \(f(x) = \sqrt{x}\) requires \(x \geq 0\), and \(g\) is defined for all real numbers, the domain of \((g \circ f)(x)\) is 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 (f∘g)(x) = f(g(x)). It requires substituting the entire output of g(x) into the function f. Understanding this process is essential to correctly form the composite functions in the problem.

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 the composite function depends on the domains of both functions and the values that keep the inner function's output within the outer function's domain.

Square Root Function Domain Restrictions

The square root function ƒ(x) = √x is only defined for x ≥ 0 because the square root of a negative number is not a real number. This restriction affects the domain of composite functions involving ƒ, requiring careful consideration of the inner function's output to ensure non-negativity.