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The function \( h(x)=\frac{1}{x+7} \) can be expressed in the form \( f(g(x)) \), where \( g(x)=(x+7) \), and \( f(x) \) is defined as: \( f(x)=\square \)

Ask by Gray Goodwin. in the United States
Feb 26,2025

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Answer

\( f(x) = \frac{1}{x} \)

Solution

We are given h(x) = 1/(x+7) and g(x) = x+7. To express h(x) as f(g(x)), we substitute g(x) into f. So, we need:   h(x) = f(g(x)) = f(x+7). Since h(x) = 1/(x+7), then we must have   f(x+7) = 1/(x+7). Letting u = x+7, we get:   f(u) = 1/u. Thus, the function f is defined by   f(x) = 1/x. So, the answer is: f(x) = 1/x.

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To express the function \( h(x) = \frac{1}{x+7} \) in the form \( f(g(x)) \), we can first identify \( g(x) = x + 7 \). Now, we need to define \( f(x) \) in such a way that when we substitute \( g(x) \) into \( f \), we get \( h(x) \). Since \( g(x) = x + 7 \), we want \( f(g(x)) = h(x) = \frac{1}{g(x)} \). Therefore, if we let \( f(x) = \frac{1}{x} \), substituting \( g(x) \) gives us: \[ f(g(x)) = f(x + 7) = \frac{1}{x + 7} = h(x). \] So, we can define \( f(x) = \frac{1}{x} \).

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