To find the inverse function of the equation:

$3x + 4y = 12$

Step 1: Express $y$ in terms of $x$

We first solve for $y$:

$4y = 12 - 3x$

$y = \frac{12 - 3x}{4}$

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$:

$x = \frac{12 - 3y}{4}$

Step 3: Solve for $y$

Multiply both sides by 4:

$4x = 12 - 3y$

Rearrange to solve for $y$:

$3y = 12 - 4x$

$y = \frac{12 - 4x}{3}$

Final Answer

$f^{-1}(x) = \frac{12 - 4x}{3}$

Would you like a step-by-step graphical explanation?

Here are some related questions:

1. How do you verify if two functions are inverses of each other?
2. How do you graph an inverse function?
3. What is the domain and range of this function and its inverse?
4. How do you find the inverse of a quadratic function?
5. What are real-life applications of inverse functions?

Tip: To check if two functions are inverses, compose them ($f(f^{-1}(x))$ and $f^{-1}(f(x))$) and see if they simplify to $x$.