Given functions f and g, find (b)(g∘ƒ)(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 83b

3장, 문제 83b

Given functions f and g, find (b)(g∘ƒ)(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 expression inside the square root must be greater than or equal to zero: $\left$(-$\frac{1}{x}$ + 2$\right$) $\geq$ 0. Also, consider the domain restrictions of g(x), which excludes x = 0 because of division by zero.

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 denominator $\sqrt{x+2}$ cannot be zero, so x + 2 > 0, which means x > -2. Also, since the square root is in the denominator, x = -2 is excluded to avoid division by 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 f(x), creating a new function that combines both operations.

Domain of a Function

The domain is the set of all input values for which a 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. Rational functions like g(x) = -1/x are undefined at x = 0, so these restrictions must be considered when determining the domain of compositions.

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