Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=√(x-1), g(x)=3x

Verified step by step guidance

1. Step 1: Understand the composition of functions. The composition (f∘g)(x) means f(g(x)).
2. Step 2: Substitute g(x) into f(x). Since g(x) = 3x, substitute 3x into f(x) to get f(3x).
3. Step 3: Evaluate f(3x). Given f(x) = \sqrt{x-1}, replace x with 3x to get \sqrt{3x-1}.
4. Step 4: Determine the domain of (f∘g)(x). The expression \sqrt{3x-1} is defined when the radicand is non-negative, so set 3x-1 \geq 0.
5. Step 5: Solve the inequality 3x-1 \geq 0 to find the domain of (f∘g)(x).

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 and then applying f to the result of g. Understanding how to correctly substitute and evaluate these functions is crucial for finding the composed function.

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 them, such as square roots or divisions by zero.

Square Root Function

The square root function, denoted as f(x) = √(x-1), is defined only for values of x that make the expression under the square root non-negative. This means x must be greater than or equal to 1. Understanding this restriction is essential for determining the overall domain of the composed function (ƒ∘g)(x).