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:

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?

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 $y = x$.