Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=x+3

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (ƒ∘g)(x) means applying g first, followed by f. For example, if g(x) = x + 3, then (ƒ∘g)(x) = f(g(x)) = f(x + 3). Understanding this process is crucial for solving the problem.

Domain of a Function

The domain of a function is the set of all possible input values (x) for which the function is defined. When composing functions, the domain of the resulting function is determined by the domain of the inner function and any restrictions imposed by the outer function. For instance, since f(x) = √x, its domain requires x to be non-negative, which affects the overall domain of (ƒ∘g)(x).

Square Root Function

The square root function, denoted as f(x) = √x, is defined only for non-negative values of x. This means that for any input to f, the output must be a real number. In the context of the problem, since g(x) = x + 3, we need to ensure that g(x) produces values that are within the domain of f, specifically that x + 3 ≥ 0, which leads to further restrictions on the domain of (ƒ∘g)(x).