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To find the inverse of the function \( g(x) = \frac{1}{x - 1} \) and determine whether the inverse is a function, let's follow these steps:

### Step 1: Find the inverse of \( g(x) \)

1. Start by writing \( y = \frac{1}{x - 1} \).
2. Swap \( x \) and \( y \) to find the inverse:
   \[
   x = \frac{1}{y - 1}
   \]
3. Solve for \( y \):
   \[
   x(y - 1) = 1
   \]
   \[
   xy - x = 1
   \]
   \[
   xy = x + 1
   \]
   \[
   y = \frac{x + 1}{x}
   \]

So, the inverse function is \( g^{-1}(x) = \frac{x + 1}{x} = 1 + \frac{1}{x} \).

### Step 2: Check if the inverse is a function

The inverse, \( g^{-1}(x) = 1 + \frac{1}{x} \), is indeed a function because each input \( x \) produces a unique output \( y \), except for \( x = 0 \), which is outside the domain.

### Conclusion

The correct choice from the given options is:

- \(\frac{1}{x} + 1, \text{ Yes}\)

Would you like more details or have any questions?

### Related Questions:

1. How do you determine the domain and range of the inverse function?
2. What are the conditions for a function to have an inverse?
3. How can you verify if an inverse function is a function using the horizontal line test?
4. What are the asymptotes of the function and its inverse?
5. How do you graph both a function and its inverse on the same coordinate plane?

**Tip:** To find the inverse of a function, swap \(x\) and \(y\) and solve for the new \(y\).

What is the inverse of g(x) = 1/(x - 1) and is it a function?

Learn how to find the inverse of g(x) = 1/(x - 1) and determine if it is a function. This problem involves key concepts in inverse functions and algebra, with step-by-step calculations and explanations. Suitable for students in Grades 10-12.

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