To find the inverse of the function $f(x) = \frac{8x + 9}{7x - 8}$, we follow these steps:

Step 1: Replace $f(x)$ with $y$

We start by writing the equation as: $y = \frac{8x + 9}{7x - 8}$

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$: $x = \frac{8y + 9}{7y - 8}$

Step 3: Solve for $y$

Now, solve for $y$ in terms of $x$. First, eliminate the denominator by multiplying both sides of the equation by $7y - 8$: $x(7y - 8) = 8y + 9$ Distribute the $x$ on the left-hand side: $7xy - 8x = 8y + 9$ Move all terms involving $y$ to one side of the equation: $7xy - 8y = 8x + 9$ Factor out $y$ on the left-hand side: $y(7x - 8) = 8x + 9$ Finally, solve for $y$ by dividing both sides by $7x - 8$: $y = \frac{8x + 9}{7x - 8}$

Thus, the inverse function is: $f^{-1}(x) = \frac{8x + 9}{7x - 8}$

Conclusion:

The function $f(x) = \frac{8x + 9}{7x - 8}$ is its own inverse. That means $f^{-1}(x) = f(x)$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. What are the conditions under which a function is its own inverse?
2. How do we determine if a function is one-to-one (bijective)?
3. Can you find the inverse of a quadratic function?
4. How can you verify that two functions are inverses of each other?
5. What is the graphical interpretation of inverse functions?

Tip: When solving for the inverse of a rational function, always ensure to check if the function is one-to-one to guarantee the existence of an inverse!