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 $x$.