To find the inverse of the function $f(x) = \frac{1}{2}x - 3$, we follow these steps:

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

$y = \frac{1}{2}x - 3$

Step 2: Swap $x$ and $y$

To find the inverse, we swap $x$ and $y$. $x = \frac{1}{2}y - 3$

Step 3: Solve for $y$

Now, solve for $y$.

1. Add 3 to both sides: $x + 3 = \frac{1}{2}y$

2. Multiply both sides by 2 to get rid of the fraction: $2(x + 3) = y$

3. Simplify: $y = 2(x + 3)$ $y = 2x + 6$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = 2x + 6$

Final Answer:

The inverse function of $f(x) = \frac{1}{2}x - 3$ is: $f^{-1}(x) = 2x + 6$

Do you need more details or have any questions? Here are 5 related questions to extend this concept:

1. How do you verify if two functions are inverses of each other?
2. What happens to the slope and intercept in the inverse function compared to the original function?
3. How can we interpret the graph of a function and its inverse geometrically?
4. What are the conditions for a function to have an inverse?
5. How does the process differ when finding the inverse of a non-linear function?

Tip: Always ensure that the original function is one-to-one (passes the horizontal line test) before finding its inverse.