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

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

Find the composition \((f \circ g)(x)\), which means \(f(g(x))\). Substitute \(g(x)\) into \(f\): \(f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x} - 2}\).

Simplify the expression inside the denominator: \(\frac{1}{\frac{1}{x} - 2} = \frac{1}{\frac{1 - 2x}{x}}\).

Rewrite the complex fraction by multiplying numerator and denominator appropriately: \(\frac{1}{\frac{1 - 2x}{x}} = \frac{1 \cdot x}{1 - 2x} = \frac{x}{1 - 2x}\).

Determine the domain of \((f \circ g)(x)\) by considering the domains of \(f\) and \(g\) and restrictions from the composition: exclude values where \(g(x)\) is undefined (i.e., \(x \neq 0\)) and where the denominator in \(f(g(x))\) is zero (i.e., \(1 - 2x \neq 0\)).

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주요 개념

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Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). It requires substituting the entire output of g(x) into the function f, 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 (f∘g)(x) includes all x-values in the domain of g for which g(x) is in the domain of f.

Rational Functions and Restrictions

Rational functions are ratios of polynomials and often have restrictions where the denominator equals zero. Identifying these restrictions is crucial to determine the domain, especially when composing functions that involve division.