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To solve this problem, let's find the inverse of the function \( f(x) = 2^{-x+5} + 4 \) and identify the lattice points for the inverse function.

### Step-by-Step Solution:

1. **Rewrite the Function as \( y \):**
   \[
   y = 2^{-x+5} + 4
   \]

2. **Isolate \( 2^{-x+5} \):**
   \[
   y - 4 = 2^{-x+5}
   \]

3. **Take the Logarithm of Both Sides to Solve for \( x \):**
   Apply the logarithm (base 2) to both sides to handle the exponent.
   \[
   -x + 5 = \log_2(y - 4)
   \]

4. **Solve for \( x \):**
   Rearrange to find \( x \) in terms of \( y \).
   \[
   x = 5 - \log_2(y - 4)
   \]

5. **Rewrite as the Inverse Function \( f^{-1}(x) \):**
   The inverse function is:
   \[
   f^{-1}(x) = 5 - \log_2(x - 4)
   \]

### Plotting Lattice Points of the Inverse

To find the lattice points, substitute some values of \( x \) and calculate \( f^{-1}(x) \) to get integer values for the coordinates. Here’s a table with some sample points:

| \( x \) | \( f^{-1}(x) = 5 - \log_2(x - 4) \) |
|---------|------------------------------------|
| \( x = 5 \) | \( y = 5 \) |
| \( x = 6 \) | \( y = 4 \) |
| \( x = 8 \) | \( y = 3 \) |
| \( x = 12 \) | \( y = 2 \) |

### Lattice Points to Plot:

- (5, 5)
- (6, 4)
- (8, 3)
- (12, 2)

These points represent the lattice points for the inverse function \( f^{-1}(x) \) and should be plotted on the graph.

Would you like more details on any specific step, or have any questions?

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Here are 5 related questions that build on this information:

1. How do you determine if a function has an inverse?
2. What transformations occur when you take the inverse of an exponential function?
3. How does the range of \( f(x) = 2^{-x+5} + 4 \) affect the domain of its inverse?
4. How would the graph of \( f(x) \) differ if the base were changed from 2 to another value?
5. How can you use logarithmic properties to simplify inverse function calculations?

**Tip:** When finding inverse functions, remember to swap \( x \) and \( y \) to make sure you're correctly reflecting the original function across the line \( y = x \).

The function f(x) = 2^{-x+5} + 4 is graphed below. Plot all lattice points of the inverse. Use the labeled points as your guide.

Learn how to find and graph the lattice points of the inverse function for f(x) = 2^{-x+5} + 4. This problem includes identifying the inverse using logarithmic functions and plotting points based on the inverse equation. Ideal for students in Grades 11-12.

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