Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \frac{2}{x}, g (x) = x + 1$

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

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

Determine the domain of $g(x)$. Since $g(x) = x + 1$ is a polynomial, its domain is all real numbers, $(-\infty, \infty)$.

Determine the domain restrictions from $f(g(x))$. Since $f(x) = \frac{2}{x}$ is undefined when the denominator is zero, set $x + 1 \neq 0$ to avoid division by zero. This gives $x \neq -1$.

Combine the domain restrictions: the domain of $(f \circ g)(x)$ is all real numbers except $x = -1$, or $(-\infty, -1) \cup (-1, \infty)$.

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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 (f∘g)(x) = f(g(x)). It means you first evaluate g(x), then use that output as the input for f. Understanding this process is essential to correctly find (f∘g)(x).

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 such that g(x) lies in the domain of f. Identifying domain restrictions is crucial to avoid undefined expressions.

Rational Functions and Restrictions

A rational function is a ratio of two polynomials, like f(x) = 2/x, which is undefined when the denominator is zero. Recognizing these restrictions helps determine the domain by excluding values that make the denominator zero, ensuring the function remains valid.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7.ƒ(x)=2x,g(x)=x+1ƒ(x)=\(\frac{2}{x}\),g(x)=x+1