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 $y = x$.