To find the inverse of the function $f(x) = 3x + 4$, follow these steps:

Step 1: Replace $f(x)$ with $y$

We start by writing the function as:

$y = 3x + 4$

Step 2: Swap $x$ and $y$

In order to find the inverse, we swap the roles of $x$ and $y$:

$x = 3y + 4$

Step 3: Solve for $y$

Now, solve for $y$ in terms of $x$.

1. Subtract 4 from both sides: $x - 4 = 3y$

2. Divide both sides by 3: $y = \frac{x - 4}{3}$

Step 4: Replace $y$ with $f^{-1}(x)$

The inverse function is:

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

Final Answer:

The inverse of $f(x) = 3x + 4$ is:

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

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Would you like further details on this process? Here are some related questions to deepen your understanding:

1. How do you find the inverse of a function when it involves more complex operations (like squaring)?
2. What is the graphical interpretation of a function and its inverse?
3. How do you check if two functions are indeed inverses of each other algebraically?
4. What is the relationship between the slopes of a function and its inverse?
5. How can you determine if a function is one-to-one, and why is that important for finding its inverse?

Tip: For linear functions, finding the inverse is often just a matter of solving for $x$ in terms of $y$. For non-linear functions, more careful consideration is needed.