Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, fg means f(g(x)), which requires evaluating g(x) first and then substituting that result into f(x). Understanding how to properly compose functions is essential for solving the problem.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative. Therefore, determining the domain involves solving inequalities to find the valid x-values for both f(x) and g(x).
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Domain Restrictions of Composed Functions
Square Root Functions
Square root functions, such as f(x) = √(x + 4) and g(x) = √(x - 1), are defined only for non-negative inputs. This means that the expressions under the square roots must be greater than or equal to zero. Understanding the behavior and restrictions of square root functions is crucial for finding their domains and composing them correctly.
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Imaginary Roots with the Square Root Property