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Let's solve this step-by-step.

### Step 1: Finding \( (f \circ g)(x) \)

Given:
\[
f(x) = \frac{1}{x} \quad \text{and} \quad g(x) = \frac{x - 1}{3}
\]

The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \):
\[
(f \circ g)(x) = f(g(x))
\]

Since \( f(x) = \frac{1}{x} \), substituting \( g(x) \) gives:
\[
(f \circ g)(x) = f\left(\frac{x - 1}{3}\right) = \frac{1}{\frac{x - 1}{3}}
\]

To simplify, we multiply by the reciprocal:
\[
(f \circ g)(x) = \frac{3}{x - 1}
\]

### Step 2: Determining the Domain of \( (f \circ g)(x) \)

The expression \( \frac{3}{x - 1} \) is undefined when the denominator \( x - 1 = 0 \), or \( x = 1 \).

Additionally, since \( g(x) = \frac{x - 1}{3} \), the input \( x \) must be defined for \( f(x) \) after substitution. Since \( f(x) = \frac{1}{x} \) requires \( g(x) \neq 0 \), we also need:
\[
\frac{x - 1}{3} \neq 0
\]
This implies \( x \neq 1 \), which is already covered by the previous restriction.

Therefore, the domain of \( (f \circ g)(x) = \frac{3}{x - 1} \) is:
\[
\text{Domain} = (-\infty, 1) \cup (1, \infty)
\]

### Final Answer:

\[
(f \circ g)(x) = \frac{3}{x - 1}
\]
\[
\text{Domain} = (-\infty, 1) \cup (1, \infty)
\]

Would you like additional details on any part? Here are related questions you might find useful:

1. What are other methods to find the domain of a composition function?
2. How does composition of functions work in general?
3. What is the difference between domain and range for composite functions?
4. Why do we need to check the domain of both functions in compositions?
5. How would the answer change if \( f(x) \) or \( g(x) \) had different forms?

**Tip:** When finding the domain of a composite function, remember to check any restrictions from each function individually before combining them.

Find the formula for (f ∘ g)(x) and simplify your answer. Then find the domain for (f ∘ g)(x). Round your answer to two decimal places.

This solution shows how to find the composition (f ∘ g)(x) for f(x) = 1/x and g(x) = (x - 1)/3. Learn the step-by-step simplification and how to determine the domain of the composite function, considering restrictions for rational functions. Ideal for high school students in Grades 10-12.

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