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.