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

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

Start by rewriting the function: $y = \frac{1}{2}x - 3$

Step 2: Swap $x$ and $y$

To find the inverse, switch $x$ and $y$: $x = \frac{1}{2}y - 3$

Step 3: Solve for $y$

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

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

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

Thus, the inverse function is: $f^{-1}(x) = 2(x + 3) \quad \text{or} \quad f^{-1}(x) = 2x + 6$

So, the inverse function in slope-intercept form is: $f^{-1}(x) = 2x + 6$

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

Related Questions:

1. How do you verify that two functions are inverses of each other?
2. What is the geometric interpretation of the inverse of a linear function?
3. How do changes in the slope or intercept of a function affect its inverse?
4. Can a function have more than one inverse? Why or why not?
5. How do you find the inverse of non-linear functions?

Tip:

When finding an inverse function, remember that switching $x$ and $y$ represents reflecting the graph across the line $y = x$.