the function (f(x)=-log_{3}(x + 3)-7) is graphed below. plot all lattice points of the inverse. use the…

Explanation:

Step1: Find the inverse of the function

Let $y =-\log_3(x + 3)-7$. First, solve for $x$ in terms of $y$. \begin{align*} y&=-\log_3(x + 3)-7\ y + 7&=-\log_3(x + 3)\ -(y + 7)&=\log_3(x + 3)\ 3^{-(y + 7)}&=x+3\ x&=3^{-(y + 7)}-3 \end{align*} So the inverse function is $f^{-1}(x)=3^{-(x + 7)}-3$.

Step2: Find lattice - points

Lattice points are points with integer coordinates. We can choose integer values for $x$ and find the corresponding integer values for $y$ (or vice - versa). Let's start by choosing some integer values of $x$. When $x=-2$, $y = 3^{-(-2 + 7)}-3=3^{-5}-3=\frac{1}{243}-3\notin\mathbb{Z}$ When $x=-3$, $y = 3^{-(-3 + 7)}-3=3^{-4}-3=\frac{1}{81}-3\notin\mathbb{Z}$ When $x = - 4$, $y=3^{-(-4 + 7)}-3=3^{-3}-3=\frac{1}{27}-3\notin\mathbb{Z}$ When $x=-10$, $y = 3^{-(-10 + 7)}-3=3^{3}-3=27 - 3=24$ When $x=-11$, $y = 3^{-(-11 + 7)}-3=3^{4}-3=81 - 3=78$ We can also work in the other direction. Let $y$ be an integer. If $y = - 6$, then $x=3^{-(-6 + 7)}-3=3^{-1}-3=\frac{1}{3}-3\notin\mathbb{Z}$ If $y=-7$, then $x=3^{-(-7 + 7)}-3=1 - 3=-2$ If $y=-8$, then $x=3^{-(-8 + 7)}-3=3 - 3=0$ The lattice points of the inverse function can be found by substituting integer values and checking for integer results. Some lattice points of $y = f^{-1}(x)$ are $(-2,-7),(0,-8),(-10,24)$ etc. Plot these points on the graph.

Answer:

Plot lattice points such as $(-2,-7),(0,-8),(-10,24)$ on the graph.