To find the inverse of the function $f(x) = 2x - 14$, follow these steps:

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

We begin by rewriting the function in terms of $y$.

$y = 2x - 14$

Step 2: Swap $x$ and $y$

Now, switch $x$ and $y$ to find the inverse function:

$x = 2y - 14$

Step 3: Solve for $y$

Next, solve for $y$:

$x + 14 = 2y$

$y = \frac{x + 14}{2}$

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

Now, write the inverse function as:

$f^{-1}(x) = \frac{x + 14}{2}$

Final Answer in slope-intercept form:

$f^{-1}(x) = \frac{1}{2}x + 7$

Would you like more details or have any questions about this process?

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Here are 5 related questions to expand your understanding:

1. How do you verify that two functions are inverses of each other?
2. What is the geometric interpretation of a function and its inverse?
3. How would you find the inverse of a quadratic function?
4. Can every function have an inverse? Why or why not?
5. What is the domain and range of $f^{-1}(x) = \frac{1}{2}x + 7$?

Tip: When finding an inverse function, always make sure the original function is one-to-one to ensure the inverse exists.