given, \quad g(x) = 2 + \frac{6}{x},  g^{-1}(x)=

To find the inverse function \( g^{-1}(x) \) for the given function \( g(x) = 2 + \frac{6}{x} \), follow these steps:

1. **Express the function in terms of \( y \)**:
   \[
   y = 2 + \frac{6}{x}
   \]

2. **Swap \( y \) and \( x \)** to find the inverse:
   \[
   x = 2 + \frac{6}{y}
   \]

3. **Solve for \( y \)**:
   - Subtract 2 from both sides:
     \[
     x - 2 = \frac{6}{y}
     \]
   - Multiply both sides by \( y \) to eliminate the fraction:
     \[
     y(x - 2) = 6
     \]
   - Solve for \( y \) by dividing both sides by \( x - 2 \):
     \[
     y = \frac{6}{x - 2}
     \]

So, the inverse function is:
\[
g^{-1}(x) = \frac{6}{x - 2}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the domain of the inverse function \( g^{-1}(x) \)?
2. What is the range of the original function \( g(x) \)?
3. How can you verify that \( g(x) \) and \( g^{-1}(x) \) are indeed inverses of each other?
4. What happens to the graph of \( g(x) \) when you apply the inverse function?
5. Can you find the inverse of a more complex function involving \( x \) in both the numerator and denominator?

**Tip:** Always check if the inverse function exists by ensuring the original function is one-to-one (passes the horizontal line test).

Learn how to find the inverse function g^-1(x) for the function g(x) = 2 + 6/x step-by-step. This problem covers the concept of inverse functions in calculus, including solving equations and verifying solutions. Suitable for high school students in Grades 10-12.

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