Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = √ (x - 1), g (x) = 3 x$

검증된 단계별 안내

Identify the given functions: $ f(x) = \sqrt{x - 1} $ and $ g(x) = 3x $.

Recall that the composition $ (g \circ f)(x) $ means $ g(f(x)) $, so substitute $ f(x) $ into $ g $.

Write the composition explicitly: $ (g \circ f)(x) = g(f(x)) = g\left( \sqrt{x - 1} \right) = 3 \times \sqrt{x - 1} $.

Determine the domain of $ (g \circ f)(x) $ by considering the domain of $ f(x) $ first, since $ f(x) $ is inside $ g $. For $ f(x) = \sqrt{x - 1} $, the expression under the square root must be $ \geq 0 $, so $ x - 1 \geq 0 $.

Solve the inequality $ x - 1 \geq 0 $ to find the domain of $ f $, which is $ x \geq 1 $. Since $ g(x) = 3x $ is defined for all real numbers, the domain of $ (g \circ f)(x) $ is the same as the domain of $ f(x) $, that is $ [1, \infty) $.

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주요 개념

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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 of g, specifically the values of x for which f(x) is in the domain of g.

Square Root Function Domain Restrictions

The square root function √(x-1) requires the expression inside the root to be non-negative, so x-1 ≥ 0. This restricts the domain of f(x) to x ≥ 1, which must be considered when determining the domain of the composite function (g∘f)(x).

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=√(x−1),g(x)=3xƒ(x)=√(x-1), g(x)=3x