Find (a) (f ∘g)(x) and the domain of f ∘g and (b) (g ∘f)(x) and the domain of g ∘f. f(x)=√(x-15)

step 1 So f composed g x equals f g x. So we just substitute x in the expression of f x with the expres

Find (a) (f ∘g)(x) and the domain of f ∘g and (b) (g ∘f)(x)...

Key Concepts

Quadratic Functions and Inequalities

Quadratic functions, such as g(x) = x² + 2x, can introduce domain restrictions when composed with functions that have additional conditions (like the square root function). To find valid inputs in a composite function, one often solves quadratic inequalities to identify the set of x values for which the composite expression remains defined. This involves factoring, finding critical points, and testing intervals.

Function Composition

Function composition involves applying one function to the result of another function. In the context of the given problem, you first calculate the inner function value and then use it as the input to the outer function. This is denoted as (f ? g)(x) = f(g(x)) or (g ? f)(x) = g(f(x)), and it requires careful attention to the order in which functions are applied.

Domain of Composite Functions

The domain of a composite function is determined by the domains of the individual functions and the way they interact. For (f ? g)(x), the input x must be in the domain of g, and additionally, g(x) must lie within the domain of f. Similarly, for (g ? f)(x), x must first satisfy the domain restrictions for f, and then f(x) must be within the domain of g. This careful analysis ensures that all expressions in the composite functions are defined.

Square Root Function

The square root function, defined as ?(expression), is only real-valued when the expression inside the square root (the radicand) is non-negative. In problems like these, it is essential to set the radicand greater than or equal to zero to determine the domain of the function. This restriction directly impacts the composite function when one of the inner functions is under the square root.

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