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, g(x)=x-1

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation (f∘g)(x) means f(g(x)), which means you substitute g(x) into f(x). Similarly, (g∘f)(x) means g(f(x)).
2. Step 2: Find (f∘g)(x). Substitute g(x) = x - 1 into f(x) = √x. So, (f∘g)(x) = f(g(x)) = f(x - 1) = √(x - 1).
3. Step 3: Determine the domain of (f∘g)(x). Since f(x) = √x, the expression inside the square root must be non-negative. Therefore, x - 1 ≥ 0, which implies x ≥ 1. The domain of (f∘g)(x) is [1, ∞).
4. Step 4: Find (g∘f)(x). Substitute f(x) = √x into g(x) = x - 1. So, (g∘f)(x) = g(f(x)) = g(√x) = √x - 1.
5. Step 5: Determine the domain of (g∘f)(x). Since f(x) = √x, x must be non-negative, so x ≥ 0. The domain of (g∘f)(x) is [0, ∞).

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 means applying g first and then f to the result. This is crucial for solving the problem as it requires understanding how to evaluate the functions in sequence.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) 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. Identifying the domain is essential to ensure valid inputs for the composed functions.

Square Root Function

The square root function, denoted as ƒ(x)=√x, is defined only for non-negative values of x, meaning x must be greater than or equal to zero. This restriction affects the domain of the composition involving this function. Understanding the properties of the square root function is vital for determining the overall domain of the composed functions in the problem.