Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x+2), g(x)=-(1/x)

Verified step by step guidance

1. Step 1: Understand the composition of functions. The composition (f∘g)(x) means you substitute g(x) into f(x), and (g∘f)(x) means you substitute f(x) into g(x).
2. Step 2: For (f∘g)(x), substitute g(x) into f(x). So, replace x in f(x) = \sqrt{x+2} with g(x) = -\frac{1}{x}.
3. Step 3: Simplify the expression for (f∘g)(x) to find the composed function.
4. Step 4: Determine the domain of (f∘g)(x). Consider the domain of g(x) and any restrictions from the square root in f(x).
5. Step 5: For (g∘f)(x), substitute f(x) into g(x). So, replace x in g(x) = -\frac{1}{x} with f(x) = \sqrt{x+2}, and determine its domain.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For functions f and g, the composition (f∘g)(x) means applying g first and then f to the result. This process is essential for solving the given problem, as it requires evaluating the functions in sequence.

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 domains of the individual functions and any restrictions imposed by their compositions. Understanding the domain is crucial for ensuring that the composed functions yield valid outputs.

Square Root and Rational Functions

The function f(x) = √(x + 2) is a square root function, which is only defined for non-negative inputs, meaning x + 2 must be greater than or equal to zero. The function g(x) = -1/x is a rational function, which is undefined when x = 0. Recognizing these characteristics is vital for determining the domains of the composed functions and ensuring valid calculations.