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We are asked to express the function \( h(x) = \frac{1}{x - 5} \) in the form \( f \circ g \), where \( g(x) = x - 5 \), and we need to find the function \( f(x) \).

In functional notation, \( f \circ g \) means \( f(g(x)) \). 

Given:
\[
h(x) = \frac{1}{x - 5}
\]
and
\[
g(x) = x - 5,
\]
we can rewrite \( h(x) \) as:
\[
h(x) = f(g(x)) = f(x - 5).
\]

Since \( h(x) = \frac{1}{x - 5} \), we compare it to \( f(x - 5) \), which implies that:
\[
f(x) = \frac{1}{x}.
\]

Thus, the function \( f(x) \) is:
\[
f(x) = \frac{1}{x}.
\]

Would you like further explanation, or do you have any additional questions?

Here are 5 related questions to expand your understanding:
1. How do we determine \( g(x) \) when decomposing a function in the form \( f \circ g \)?
2. Can you decompose other functions like \( h(x) = \frac{1}{x+2} \) in a similar way?
3. What is the general process to express any given function as a composition of two functions?
4. How would \( f \) and \( g \) change if \( h(x) = \frac{2}{x-3} \)?
5. What are the properties of compositions of functions, and how do they affect the final expression?

**Tip**: When decomposing a function, always start by identifying the inner function \( g(x) \), which is applied first, and then identify the outer function \( f(x) \), which is applied to the result of \( g(x) \).

Express the function h(x) = 1 / (x - 5) in the form f ∘ g. If g(x) = x - 5, find the function f(x).

Learn how to express the function h(x) = 1 / (x - 5) in the form f ∘ g. This problem covers key algebraic concepts related to function composition. Suitable for students in Grades 9-12.

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