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To find the inverse of the function \( f(x) = -\frac{1}{4}x + 2 \), let's follow these steps:

1. **Rewrite the function as \( y \):**
   \[
   y = -\frac{1}{4}x + 2
   \]

2. **Swap \( x \) and \( y \):**
   This is done because finding the inverse means finding the values of \( y \) that correspond to values of \( x \) in the original function.
   \[
   x = -\frac{1}{4}y + 2
   \]

3. **Solve for \( y \):**
   Isolate \( y \) on one side of the equation.
   - Subtract 2 from both sides:
     \[
     x - 2 = -\frac{1}{4}y
     \]
   - Multiply both sides by \(-4\) to clear the fraction:
     \[
     -4(x - 2) = y
     \]
   - Distribute the \(-4\):
     \[
     y = -4x + 8
     \]

4. **Rewrite the inverse function:**
   Now, we have \( y = -4x + 8 \), which is the inverse function in slope-intercept form.

So, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = -4x + 8
\]

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

### Related Questions:
1. How do you verify that two functions are inverses of each other?
2. What is the importance of switching \( x \) and \( y \) in finding the inverse of a function?
3. How would you find the inverse of a quadratic function?
4. Can all functions have inverses? Why or why not?
5. What does the slope of the inverse function tell us about the original function?

### Tip:
To check if two functions are inverses, compose them (i.e., \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \)) and see if both yield \( x \).

Find the inverse function in slope-intercept form (mx + b) for f(x) = -1/4 x + 2.

Learn how to find the inverse of the linear function f(x) = -1/4x + 2 in slope-intercept form. This solution explains step-by-step methods for finding inverses, isolating variables, and simplifying linear equations. Suitable for high school students in grades 9-12.

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