Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=1/(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 f(g(x)).
2. Step 2: Substitute g(x) into f(x). Since g(x) = 1/x, substitute this into f(x) to get f(g(x)) = f(1/x).
3. Step 3: Substitute 1/x into f(x). The function f(x) = 1/(x-2) becomes f(1/x) = 1/((1/x) - 2).
4. Step 4: Simplify the expression. Simplify the expression 1/((1/x) - 2) by finding a common denominator for the terms in the denominator.
5. Step 5: Determine the domain. Consider the restrictions from both f(x) and g(x). For g(x), x cannot be 0. For f(x), the expression inside the denominator cannot be zero. Solve (1/x) - 2 ≠ 0 to find additional restrictions.

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. 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) 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 that arise from their definitions, such as division by zero.

Identifying Restrictions

Identifying restrictions is essential when dealing with rational functions, as certain values can make the function undefined. For the given functions f(x) = 1/(x-2) and g(x) = 1/x, we must consider values that would lead to division by zero, which will affect both the domain of the composed function and the overall solution.