Question
he function \( h(x)=\frac{1}{x-5} \) can be expressed in the form \( f(g(x)) \), where \( g(x)=(x-5) \), and \( f(x) \) is lefined as: \( f(x)=\square \)
Ask by Matthews Howell. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x) = \frac{1}{x} \)
Solution
Given the function
\[
h(x)=\frac{1}{x-5},
\]
we want to express it as
\[
h(x)=f(g(x))
\]
with
\[
g(x)=x-5.
\]
Since
\[
h(x)=\frac{1}{x-5} \quad \text{and} \quad g(x)=x-5,
\]
we substitute \( g(x) \) into \( f \) such that:
\[
f(g(x))=\frac{1}{g(x)}.
\]
This means that the function \( f(x) \) must be defined as:
\[
f(x)=\frac{1}{x}.
\]
Thus, the answer is:
\[
f(x)=\frac{1}{x}.
\]
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Simplify this solution Mind Expander
To express the function \( h(x) = \frac{1}{x-5} \) in the form \( f(g(x)) \), we start with \( g(x) = x - 5 \). The function \( f(x) \) is what transforms \( g(x) \) to give \( h(x) \). Since \( h(x) \) is the reciprocal of \( g(x) \), we need \( f(x) \) to be the reciprocal function. Thus, we define: \( f(x) = \frac{1}{x} \). Putting it all together, we have \( h(x) = f(g(x)) = f(x-5) = \frac{1}{x-5} \). Now, we’ve successfully expressed \( h(x) \) as a composition of two functions!
