To find the inverse of the function $f(x) = 2(x + 4)$, follow these steps:

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

Write the function as: $y = 2(x + 4)$

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$: $x = 2(y + 4)$

Step 3: Solve for $y$

Now solve the equation for $y$.

1. Divide both sides by 2 to isolate the $y + 4$ term: $\frac{x}{2} = y + 4$

2. Subtract 4 from both sides: $y = \frac{x}{2} - 4$

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

The inverse function is: $f^{-1}(x) = \frac{x}{2} - 4$

Final Answer:

$f^{-1}(x) = \frac{x}{2} - 4$

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Would you like further explanation on any of the steps? Here are some related questions you might find interesting:

1. What does it mean geometrically for a function and its inverse?
2. How can you verify that two functions are inverses of each other?
3. What is the domain and range of the function $f(x) = 2(x + 4)$?
4. How do you graph a function and its inverse?
5. What happens if a function is not one-to-one?

Tip: Always remember that for a function to have an inverse, it must be one-to-one (meaning each $x$-value corresponds to exactly one $y$-value).