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.