Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

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Identify the composition of functions (g \(\circ\) f)(x), which means g(f(x)). This involves substituting the entire function f(x) into g(x).

Write down the given functions explicitly: \( f(x) = \frac{1}{x-2} \) and \( g(x) = \frac{1}{x} \).

Substitute \( f(x) \) into \( g(x) \) to get \( (g \circ f)(x) = g\left( \frac{1}{x-2} \right) = \frac{1}{\frac{1}{x-2}} \).

Simplify the expression \( \frac{1}{\frac{1}{x-2}} \) by multiplying numerator and denominator appropriately to find the simplified form of \( (g \circ f)(x) \).

Determine the domain of \( (g \circ f)(x) \) by considering the restrictions from both \( f(x) \) and \( g(x) \): exclude values where \( x-2=0 \) (to avoid division by zero in \( f(x) \)) and where \( f(x) = 0 \) (to avoid division by zero in \( g(f(x)) \)).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘f)(x) = g(f(x)). It requires substituting the entire output of f(x) into g, creating a new function that combines both operations.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When composing functions, the domain of (g∘f)(x) includes all x-values in the domain of f for which f(x) is in the domain of g.

Restrictions from Rational Functions

Rational functions like f(x) = 1/(x-2) and g(x) = 1/x have restrictions where denominators cannot be zero. Identifying these values is essential to determine the domain and ensure the function is defined.