To find the inverse function of $f(x) = \frac{7x - 5}{8x + 1}$, follow these steps:

Step 1: Set $y = f(x)$

This gives us: $y = \frac{7x - 5}{8x + 1}$

Step 2: Solve for $x$

Now, express $x$ in terms of $y$.

Multiply both sides by $8x + 1$ to eliminate the denominator: $y(8x + 1) = 7x - 5$

Distribute $y$ on the left-hand side: $8xy + y = 7x - 5$

Move all terms involving $x$ to one side and the constant terms to the other: $8xy - 7x = -5 - y$

Factor out $x$ from the left-hand side: $x(8y - 7) = -5 - y$

Now solve for $x$: $x = \frac{-5 - y}{8y - 7}$

Step 3: Replace $y$ with $x$ to express the inverse

Thus, the inverse function is: $f^{-1}(x) = \frac{-5 - x}{8x - 7}$

Step 4: Check using the Inverse Function Property

To verify that this is indeed the inverse function, we need to check whether:

1. $f(f^{-1}(x)) = x$
2. $f^{-1}(f(x)) = x$

Check $f(f^{-1}(x))$:

Substitute $f^{-1}(x)$ into $f(x)$: $f(f^{-1}(x)) = \frac{7\left(\frac{-5 - x}{8x - 7}\right) - 5}{8\left(\frac{-5 - x}{8x - 7}\right) + 1}$ Simplify this expression: $f(f^{-1}(x)) = x$ This shows that $f(f^{-1}(x)) = x$.

Check $f^{-1}(f(x))$:

Substitute $f(x)$ into $f^{-1}(x)$: $f^{-1}(f(x)) = \frac{-5 - \left(\frac{7x - 5}{8x + 1}\right)}{8\left(\frac{7x - 5}{8x + 1}\right) - 7}$ Simplify this expression: $f^{-1}(f(x)) = x$ This confirms that $f^{-1}(f(x)) = x$.

Thus, the inverse function is correct.

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Would you like further details on any step or have questions about this?
Here are some related questions you might find useful:

1. How do you find the domain and range of a rational function?
2. What is the general process of finding an inverse for non-linear functions?
3. Can every function have an inverse? Why or why not?
4. How do you graph a function and its inverse?
5. How can you prove that a function is one-to-one?

Tip: Always check if a function is one-to-one before finding its inverse by using the Horizontal Line Test!