Let and . Find each of the following, if possible.
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Understand that the composition of functions (g \(\circ\) ƒ)(x) means you substitute the output of ƒ(x) into g(x). In other words, (g \(\circ\) ƒ)(x) = g(ƒ(x)).
Identify the given functions: ƒ(x) = \(\sqrt{x - 2}\) and g(x) = x^2.
Substitute ƒ(x) into g(x): replace every x in g(x) with \(\sqrt{x - 2}\), so g(ƒ(x)) = (\(\sqrt{x - 2}\))^2.
Simplify the expression (\(\sqrt{x - 2}\))^2 by using the property that squaring a square root returns the original expression inside the root, so it simplifies to x - 2.
Determine the domain of the composition (g \(\circ\) ƒ)(x) by considering the domain of ƒ(x), which requires x - 2 \(\geq\) 0, so x \(\geq\) 2.
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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 requires substituting the entire output of the inner function into the outer function. Understanding this process is essential to correctly evaluate composite functions.
The domain is the set of all input values for which a function is defined. When composing functions, the domain of the composite function depends on the domain of the inner function and the domain restrictions of the outer function after substitution. Identifying these restrictions ensures valid inputs.
The function ƒ(x) = √(x-2) is a square root function, defined only for x ≥ 2 to keep the expression under the root non-negative. The function g(x) = x² is a quadratic function, defined for all real numbers. Knowing their properties helps in evaluating and composing these functions correctly.