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

### Step 1: Rewrite the function
We have the function:
\[
f(x) = -\frac{1}{3}x + 1
\]
This can also be written as:
\[
y = -\frac{1}{3}x + 1
\]

### Step 2: Swap \(x\) and \(y\)
To find the inverse, switch the roles of \(x\) and \(y\):
\[
x = -\frac{1}{3}y + 1
\]

### Step 3: Solve for \(y\)
Now, solve for \(y\):
1. Subtract 1 from both sides:
   \[
   x - 1 = -\frac{1}{3}y
   \]
2. Multiply both sides by \(-3\) to isolate \(y\):
   \[
   -3(x - 1) = y
   \]
   Simplify:
   \[
   y = -3x + 3
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = -3x + 3
\]

### Answer
The correct answer is **A**: \( f^{-1}(x) = -3x + 3 \).

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Do you want further details or have any questions? Here are five related questions to explore further:

1. How do you verify if two functions are inverses of each other?
2. What is the geometric meaning of finding the inverse of a linear function?
3. How does the slope of the original function relate to the inverse?
4. Can every function have an inverse? Why or why not?
5. How would the inverse change if the function included quadratic or higher-degree terms?

**Tip**: When finding the inverse of a function, always make sure the original function is one-to-one, so that the inverse exists.

Learn how to find the inverse of the linear function f(x) = -(1/3)x + 1. This step-by-step solution demonstrates how to switch variables and solve for y, which results in the inverse f⁻¹(x) = -3x + 3. Suitable for students in Grades 9-11.

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