Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√(x-2), g(x)=2x

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Identify the given functions: $f(x) = \sqrt{x - 2}$ and $g(x) = 2x$.

To find $(f \circ g)(x)$, substitute $g(x)$ into $f$: write $f(g(x)) = f(2x) = \sqrt{2x - 2}$.

Determine the domain of $(f \circ g)(x)$ by setting the expression inside the square root to be greater than or equal to zero: $2x - 2 \geq 0$. Solve this inequality for $x$.

To find $(g \circ f)(x)$, substitute $f(x)$ into $g$: write $g(f(x)) = g(\sqrt{x - 2}) = 2 \cdot \sqrt{x - 2}$.

Determine the domain of $(g \circ f)(x)$ by considering the domain of $f(x)$ first, since $g$ is defined for all real numbers. Set the inside of the square root $x - 2 \geq 0$ and solve for $x$.

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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 (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 depends on the domain of the inner function and the domain restrictions of the outer function after substitution. Identifying these restrictions ensures the composite function is valid.

Square Root Function Restrictions

The square root function √(x) is only defined for x ≥ 0. For f(x) = √(x - 2), the expression inside the root must be non-negative, so x - 2 ≥ 0, or x ≥ 2. This restriction affects the domain of the composite functions involving f, as the input to f must satisfy this condition.

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