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We are tasked with selecting the graph of the inverse of the function \( f(x) = -2x + 4 \).

### Step-by-Step Process:

1. **Original Function**: The function given is \( f(x) = -2x + 4 \). This is a linear function with a slope of -2 and a y-intercept of 4.
   
2. **Inverse Function**:
   To find the inverse of \( f(x) = -2x + 4 \):
   - Replace \( f(x) \) with \( y \): 
     \[
     y = -2x + 4
     \]
   - Swap \( x \) and \( y \): 
     \[
     x = -2y + 4
     \]
   - Solve for \( y \):
     \[
     x - 4 = -2y
     \]
     \[
     y = \frac{4 - x}{2}
     \]
     or equivalently: 
     \[
     y = 2 - \frac{x}{2}
     \]
     This is the equation of the inverse function \( f^{-1}(x) = 2 - \frac{x}{2} \), which has a slope of \( -\frac{1}{2} \) and a y-intercept of 2.

3. **Graphing the Inverse**: 
   - The inverse function \( f^{-1}(x) \) should have a line with a slope of \( -\frac{1}{2} \), slanting downwards but less steep than the original function.
   - Its y-intercept is at \( y = 2 \).

### Matching the Graphs:
- The correct graph should show a line with a y-intercept of 2 and a slope that descends less steeply compared to the original.

From the provided options:
- **Graph A**: This has a negative slope, but it looks too steep to match \( f^{-1}(x) \).
- **Graph B**: This has a positive slope, which is incorrect since the inverse has a negative slope.
- **Graph C**: This shows a line with a negative slope that matches the inverse function’s slope and has a y-intercept of 2.

### Conclusion:
The correct graph is **C**.

Would you like any further details on this problem, or do you have any other questions?

Here are five related questions to deepen your understanding:
1. How do you find the inverse of a linear function algebraically?
2. What are the geometric properties of inverse functions on a graph?
3. How does the slope of a line affect the steepness of a graph?
4. Why is the line of an inverse function reflected over the line \( y = x \)?
5. Can every function have an inverse? Why or why not?

**Tip**: When graphing an inverse function, always remember that it reflects the original function across the line \( y = x \).

Select the graph of the inverse of the function f(x) = -2x + 4.

Learn how to graph the inverse of the linear function f(x) = -2x + 4. This problem involves key algebraic concepts such as finding inverse functions and graphing them with specific slopes and intercepts. Suitable for high school students in Grades 9-11.

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