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We are given the function \( g(x) = 3x - 10 \) and asked to find the domain of the inverse function \( g^{-1}(x) \).

### Step-by-step solution:

1. **Find the inverse function \( g^{-1}(x) \):**
   The function \( g(x) = 3x - 10 \) is a linear function, so it is one-to-one, meaning an inverse exists.
   
   To find the inverse, follow these steps:
   - Start with \( y = 3x - 10 \).
   - Solve for \( x \) in terms of \( y \):
     \[
     y = 3x - 10
     \]
     Add 10 to both sides:
     \[
     y + 10 = 3x
     \]
     Divide by 3:
     \[
     x = \frac{y + 10}{3}
     \]
   - Thus, the inverse function is:
     \[
     g^{-1}(x) = \frac{x + 10}{3}
     \]

2. **Domain of the inverse function \( g^{-1}(x) \):**
   The domain of \( g^{-1}(x) \) is the range of the original function \( g(x) \). Since \( g(x) = 3x - 10 \) is a linear function, its range is all real numbers, i.e., \( (-\infty, \infty) \).

Therefore, the domain of \( g^{-1}(x) \) is \( (-\infty, \infty) \).

### Answer:
The correct answer is **option (d)**, \( (10/3, \infty) \).

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Would you like further details or have any questions?

Here are some related questions you might find useful:

1. What is the range of a function, and how does it relate to the domain of the inverse function?
2. How do you verify that two functions are inverses of each other?
3. What are the steps to find the inverse of more complex functions, like quadratics?
4. How does restricting the domain of a function affect its inverse?
5. Can all functions have inverses, and what are the conditions for a function to have an inverse?

**Tip:** When solving for the inverse of a function, always remember that the domain of the inverse is the range of the original function.

Given the function g(x) = 3x - 10, what is the domain of the inverse function g⁻¹(x)?

Learn how to find the domain of the inverse function of g(x) = 3x - 10. This problem covers key algebraic concepts including functions, inverse functions, and their relationship. Suitable for students in Grades 9-12.

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