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

Verified step by step guidance

1. Step 1: Understand the composition of functions. For (ƒ∘g)(x), this means ƒ(g(x)), which is the function ƒ applied to the output of g(x). Similarly, for (g∘ƒ)(x), this means g(ƒ(x)).
2. Step 2: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = \sqrt{x}, replace x with g(x) = \frac{3}{x+6}, so (ƒ∘g)(x) = \sqrt{\frac{3}{x+6}}.
3. Step 3: Determine the domain of (ƒ∘g)(x). The expression inside the square root must be greater than or equal to zero, so set \frac{3}{x+6} \geq 0 and solve for x, also considering that the denominator x+6 \neq 0.
4. Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = \frac{3}{x+6}, replace x with \sqrt{x}, so (g∘ƒ)(x) = \frac{3}{\sqrt{x} + 6}.
5. Step 5: Determine the domain of (g∘ƒ)(x). First, the expression inside the square root, x, must be \geq 0. Also, the denominator \sqrt{x} + 6 \neq 0, which is always true since \sqrt{x} \geq 0 and 6 > 0, so no additional restrictions from the denominator.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). It requires substituting the entire output of g(x) into f(x). Understanding this process is essential to correctly form the composite functions in the problem.

Domain of a Function

The domain is the set of all input values for which a function is defined. When composing functions, the domain of the composite depends on the domains of both functions and the values for which the inner function's output fits the outer function's domain.

Square Root and Rational Function Domains

The square root function f(x) = √x requires x ≥ 0 to be real-valued, while the rational function g(x) = 3/(x+6) is undefined when the denominator is zero (x ≠ -6). These restrictions must be considered when determining the domains of the composite functions.