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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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).

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Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.