Question
The function \( h(x)=\frac{1}{x-6} \) can be expressed in the form \( f(g(x)) \), where \( g(x)=(x-6) \), and \( f(x) \) is defined as: \( f(x)= \)
Ask by Reed Rodriguez. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x) = \frac{1}{x} \)
Solution
1. We start with the function \( h(x)=\frac{1}{x-6} \).
2. We are given the decomposition \( h(x)=f(g(x)) \) where \( g(x)=x-6 \).
3. To express \( h(x) \) in the form \( f(g(x)) \), let
\[
u=g(x)=x-6.
\]
Then, \( h(x) \) becomes:
\[
h(x)=\frac{1}{u}.
\]
4. This shows that the outer function \( f(u) \) must satisfy:
\[
f(u)=\frac{1}{u}.
\]
By replacing \( u \) with \( x \) (since it represents the input to \( f \)), we have:
\[
f(x)=\frac{1}{x}.
\]
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Simplify this solution The Deep Dive
To express the function \( h(x) = \frac{1}{x-6} \) in the form \( f(g(x)) \), we first identify \( g(x) = x - 6 \). To find the function \( f(x) \), we need to substitute \( g(x) \) into \( h(x) \). Since \( h(x) = \frac{1}{g(x)} \), we can deduce that \( f(x) = \frac{1}{x} \). Therefore, \( f(x) \) is defined as: \[ f(x) = \frac{1}{x} \] With this, we can rewrite \( h(x) \) as \( h(x) = f(g(x)) = f(x-6) = \frac{1}{x-6} \). Now you have distinct functions \( f \) and \( g \) working together to form \( h \)!
