To find the inverse function of $f(x) = 3x - 12$, we can proceed as follows:

Step 1: Rewrite $f(x)$ as $y$

$y = 3x - 12$

Step 2: Swap $x$ and $y$

This gives us: $x = 3y - 12$

Step 3: Solve for $y$

To isolate $y$, add 12 to both sides: $x + 12 = 3y$

Then, divide by 3: $y = \frac{x + 12}{3}$

Step 4: Write the inverse function notation

The inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{x + 12}{3}$

Final Answer in Slope-Intercept Form

If we rewrite this in slope-intercept form $y = mx + b$: $f^{-1}(x) = \frac{1}{3}x + 4$

So, the inverse function is: $f^{-1}(x) = \frac{1}{3}x + 4$

Would you like further details on this solution, or do you have any questions?

Related Questions

1. How do we verify if two functions are inverses of each other?
2. What is the domain and range of the original function $f(x) = 3x - 12$?
3. How do inverse functions relate to reflections over the line $y = x$?
4. How would we find the inverse if the function had a different slope, such as $f(x) = 5x - 12$?
5. Can every function have an inverse? Why or why not?

Tip

When finding an inverse function, always check that the original function is one-to-one to ensure the inverse exists and is also a function.