To find the inverse of the given function $f(x) = 6x$, follow these steps:

1. Start with the equation: $y = 6x$

2. Swap $x$ and $y$ to find the inverse: $x = 6y$

3. Solve for $y$: $y = \frac{x}{6}$

4. Express the inverse function: $f^{-1}(x) = \frac{x}{6}$

Final Answer:

The inverse function is: $f^{-1}(x) = \frac{x}{6}$

Would you like a detailed explanation of any step? Let me know!

Related Questions:

1. What is the domain and range of $f(x) = 6x$ and its inverse?
2. How do we verify that two functions are inverses of each other?
3. What happens to the slope when finding the inverse of a linear function?
4. Can every function have an inverse? Why or why not?
5. How would the process differ for a quadratic function, such as $f(x) = x^2$?

Tip:

When finding the inverse of a function, always ensure that the original function is one-to-one (i.e., passes the horizontal line test). This guarantees the existence of an inverse!