Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \frac{4}{x}, g (x) = x + 4$

검증된 단계별 안내

Step 1: Understand the composition of functions. The composition (ƒ∘g)(x) means ƒ(g(x)), which is applying g first, then ƒ to the result. Similarly, (g∘ƒ)(x) means g(ƒ(x)).

Step 2: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = 4/x and g(x) = x + 4, write (ƒ∘g)(x) = ƒ(g(x)) = 4 / (x + 4).

Step 3: Determine the domain of (ƒ∘g)(x). The domain consists of all x-values for which the expression is defined. Since the denominator cannot be zero, set x + 4 ≠ 0 and solve for x.

Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = x + 4 and ƒ(x) = 4/x, write (g∘ƒ)(x) = g(ƒ(x)) = (4/x) + 4.

Step 5: Determine the domain of (g∘ƒ)(x). The domain consists of all x-values for which ƒ(x) is defined (since it is inside g). Since ƒ(x) = 4/x, x cannot be zero.

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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 requires substituting the entire output of g(x) into the function f. Understanding this process is essential to correctly form the composite functions in the problem.

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 the composite function depends on the domains of both functions and the values for which the inner function's output lies within the domain of the outer function.

Rational Functions and Restrictions

A rational function is a ratio of polynomials, such as f(x) = 4/x, which is undefined when the denominator is zero. Identifying values that make the denominator zero is crucial to determine domain restrictions, especially when these functions are part of compositions.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=4x,g(x)=x+4ƒ(x)=\(\frac{4}{x}\),g(x)=x+4