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The function given is \( f(x) = -\log_2(-(x-2)) - 2 \), and you are asked to plot lattice points of the inverse and provide information about the inverse function, including asymptotes, range, and behavior.

### Step 1: Find the inverse function
To find the inverse \( f^{-1}(x) \), follow these steps:

1. Set \( y = f(x) = -\log_2(-(x-2)) - 2 \).
2. Solve for \( x \) in terms of \( y \).

\[
y = -\log_2(-(x - 2)) - 2
\]
Add 2 to both sides:
\[
y + 2 = -\log_2(-(x - 2))
\]
Multiply both sides by -1:
\[
-(y + 2) = \log_2(-(x - 2))
\]
Rewrite the equation in exponential form:
\[
2^{-(y + 2)} = -(x - 2)
\]
Now solve for \( x \):
\[
x - 2 = -2^{-(y + 2)}
\]
\[
x = 2 - 2^{-(y + 2)}
\]
Thus, the inverse function is:
\[
f^{-1}(x) = 2 - 2^{-(x + 2)}
\]

### Step 2: Identify asymptotes and behavior
- **Horizontal asymptote:** The inverse function has a horizontal asymptote at \( y = 2 \), because as \( x \to \infty \), \( 2^{-(x+2)} \) approaches 0.
- **Range of the inverse function:** Since the original function \( f(x) \) has a range of \( (-\infty, -2) \), the inverse function \( f^{-1}(x) \) will have a **domain** of \( (-\infty, -2) \) and a **range** of \( (-\infty, 2) \).
- **Behavior:** The inverse is decreasing on its domain, as shown by the negative power of 2 in the inverse expression.

### Step 3: Lattice points of the inverse
The inverse is a transformation of the original function, so you should plot points where the coordinates switch, reflecting across the line \( y = x \). For example, if the original function has a point \( (x_1, y_1) \), the inverse function will have the point \( (y_1, x_1) \).

### Answer:
The inverse function is:
\[
f^{-1}(x) = 2 - 2^{-(x+2)}
\]
- It has a **horizontal** asymptote at \( y = 2 \).
- The range of the inverse function is \( (-\infty, 2) \), and it is **decreasing** on its domain \( (-\infty, -2) \).

Would you like further details or explanations?

---

Here are some follow-up questions to extend your understanding:
1. How does the base of the logarithm affect the shape of the graph of \( f(x) \)?
2. What would happen if the logarithmic function had a positive coefficient instead of negative?
3. Can you find the derivative of \( f^{-1}(x) \) using the inverse function rule?
4. How do you identify lattice points for more complex inverses?
5. How does the domain of \( f(x) \) change when shifting the function horizontally?

**Tip:** Always remember that the domain of the inverse function is the range of the original function and vice versa.

The function f(x) = -log_2(-(x - 2)) - 2 is graphed below. Plot all lattice points of the inverse. Use the labeled points as your guide.

Learn how to find and graph the inverse of the logarithmic function f(x) = -log_2(-(x - 2)) - 2. This includes identifying the inverse function, understanding its horizontal asymptote, and plotting lattice points. Suitable for students in Grades 10-12.

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