To find the inverse function of $f(x) = \frac{7x}{5}$, let's follow these steps:

1. Replace $f(x)$ with $y$: $y = \frac{7x}{5}$

2. Swap $x$ and $y$: $x = \frac{7y}{5}$

3. Solve for $y$: To solve for $y$, multiply both sides by $\frac{5}{7}$: $y = \frac{5}{7}x$

4. Write the inverse function: The inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{5}{7}x$

So, the inverse function of $f(x) = \frac{7x}{5}$ is:

$f^{-1}(x) = \frac{5}{7}x$

Would you like any further details or have any questions? Here are some related questions to explore:

1. How do you verify that $f$ and $f^{-1}$ are indeed inverses?
2. What is the inverse of a function if it involves both multiplication and addition, like $f(x) = 2x + 3$?
3. Can every function have an inverse? Why or why not?
4. How do you find the inverse of a quadratic function?
5. What does it mean geometrically for two functions to be inverses?

Tip: The graph of a function and its inverse are reflections of each other over the line $y = x$.