Question
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
Upstudy AI Solution
Tutor-Verified Answer
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.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Beyond the Answer
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 \( f(x) = \frac{1}{x} \).