Find the inverse function in slope-intercept form (mx+b): f(x)=1/2x-3

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 \( y = x \).

Learn how to find the inverse function of f(x) = 1/2x - 3 in slope-intercept form. This step-by-step guide covers essential algebra concepts, including the inverse function process, linear equations, and solving for y. Suitable for students in Grades 9-11.

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