For each pair of functions, determine a) (f ∘g)(x), b)(f...

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Key Concepts

Domain Considerations in Function Composition

When composing functions, it's important to consider the domains of the functions involved. The composite function's domain consists of those inputs x that not only belong to the domain of the inner function but also produce outputs that lie within the domain of the outer function. This is particularly vital when dealing with functions that have restrictions, such as square root functions where the radicand must be non-negative.

Substitution in Functions

Substitution involves replacing a variable with an expression. When evaluating a composite function such as f(g(x)) or g(f(x)), the output of the inner function is substituted into the outer function in place of the variable. This technique is fundamental to correctly compute the composite function and determine its value at specific inputs.

Function Composition

Function composition is the process of applying one function to the results of another. For two functions, f and g, the composition f ? g is defined as f(g(x)), which means that you first apply the function g to x and then apply the function f to the result of g(x). The order of composition is crucial because generally f ? g is not the same as g ? f.