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The problem asks to find \( g^{-1}(6) \), the inverse of the function \( g(x) = \frac{3}{x + 4} \) evaluated at 6.

### Step-by-step solution:

1. **Set \( g(x) \) equal to 6**:
   \[
   g(x) = \frac{3}{x + 4}
   \]
   To find \( g^{-1}(6) \), we set:
   \[
   \frac{3}{x + 4} = 6
   \]

2. **Solve for \( x \)**:
   Multiply both sides of the equation by \( (x + 4) \):
   \[
   3 = 6(x + 4)
   \]
   Now, divide by 6:
   \[
   \frac{3}{6} = x + 4 \implies \frac{1}{2} = x + 4
   \]
   Subtract 4 from both sides:
   \[
   x = \frac{1}{2} - 4 = -\frac{7}{2}
   \]

Thus, \( g^{-1}(6) = -\frac{7}{2} \).

Would you like further details on how this solution works or have any additional questions?

### Related questions:
1. How do you find the inverse of a function in general?
2. Can you explain the concept of function composition with inverses?
3. What is the domain and range of \( g(x) = \frac{3}{x+4} \)?
4. How would you find \( g^{-1}(y) \) for any arbitrary value \( y \)?
5. Can you graph both \( g(x) \) and \( g^{-1}(x) \) to visualize their relationship?

### Tip:
Always check that the inverse function satisfies the condition \( g(g^{-1}(x)) = x \) when solving for inverses.

Find g^(-1)(6) given that g(x) = 3 / (x + 4)

Learn how to find the inverse function g^(-1)(6) given g(x) = 3 / (x + 4) with this step-by-step solution. This algebra problem focuses on rational functions and inverse function properties, suitable for students in Grades 9-12.

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