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

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 83a

3장, 문제 83a

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

검증된 단계별 안내

Step 1: Understand the composition of functions. For (ƒ∘g)(x), this means ƒ(g(x)), which is the function ƒ applied to the output of g(x). Similarly, for (g∘ƒ)(x), this means g(ƒ(x)).

Step 2: Find (ƒ∘g)(x) by substituting g(x) into ƒ. Since ƒ(x) = $\sqrt{x+2}$, replace x with g(x) = -$\frac{1}{x}$ to get (ƒ∘g)(x) = $\sqrt{-\frac{1}{x}$ + 2}.

Step 3: Determine the domain of (ƒ∘g)(x). The domain must satisfy two conditions: (a) the domain of g(x), which excludes x = 0 because of division by zero, and (b) the expression inside the square root in ƒ(g(x)) must be greater than or equal to zero, so $\left$(-$\frac{1}{x}$ + 2$\right$) $\geq$ 0.

Step 4: Find (g∘ƒ)(x) by substituting ƒ(x) into g. Since g(x) = -$\frac{1}{x}$, replace x with ƒ(x) = $\sqrt{x+2}$ to get (g∘ƒ)(x) = -$\frac{1}{\sqrt{x+2}$}.

Step 5: Determine the domain of (g∘ƒ)(x). The domain must satisfy: (a) the domain of ƒ(x), which requires the radicand $\left$(x+2$\right$) $\geq$ 0, and (b) the denominator in g(ƒ(x)) cannot be zero, so $\sqrt{x+2}$ $\neq$ 0, which means $\left$(x+2$\right$) > 0.

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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 f(x), creating a new function that combines both operations sequentially.

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 domain of the inner function and the domain restrictions of the outer function after substitution.

Square Root and Rational Function Restrictions

The square root function requires the radicand to be non-negative, limiting inputs to values where x+2 ≥ 0. The rational function 1/x is undefined at x=0, so its domain excludes zero. These restrictions must be considered when finding domains of composed functions.

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