To find the inverse function of $y = \frac{2x + 1}{x - 5}$, we proceed with the following steps:

Step 1: Rewrite the Function

The function given is: $y = \frac{2x + 1}{x - 5}$

To find the inverse, we’ll first replace $y$ with $x$ and $x$ with $y$: $x = \frac{2y + 1}{y - 5}$

Step 2: Solve for $y$ in Terms of $x$

Now, we want to isolate $y$ on one side of the equation.

1. Start by multiplying both sides by $y - 5$ to get rid of the denominator: $x(y - 5) = 2y + 1$

2. Distribute $x$ on the left side: $xy - 5x = 2y + 1$

3. Move terms involving $y$ to one side and constants to the other: $xy - 2y = 5x + 1$

4. Factor out $y$ from the left side: $y(x - 2) = 5x + 1$

5. Finally, solve for $y$ by dividing both sides by $x - 2$: $y = \frac{5x + 1}{x - 2}$

Conclusion

The inverse function of $y = \frac{2x + 1}{x - 5}$ is: $y^{-1} = \frac{5x + 1}{x - 2}$

Would you like additional details or have any questions?

Here are some related questions to consider:

1. How can we verify if two functions are inverses of each other?
2. What is the domain and range of the inverse function $y^{-1} = \frac{5x + 1}{x - 2}$?
3. How does the process change if we have a different type of function, such as a quadratic?
4. What applications in real life involve finding inverse functions?
5. How do we handle inverse functions for functions that are not one-to-one?

Tip: Always check that the function is one-to-one before attempting to find its inverse; this ensures that an inverse function exists.