the function $f(x)=log_3(x + 3)-7$ is graphed below. plot all lattice points of the inverse. use the labeled…

the function $f(x)=log_3(x + 3)-7$ is graphed below. plot all lattice points of the inverse. use the labeled points as your guide. click on the graph to plot a point. click a point to delete it.

the function $f(x)=log_3(x + 3)-7$ is graphed below. plot all lattice points of the inverse. use the labeled points as your guide. click on the graph to plot a point. click a point to delete it.

Answer

Explanation:

Step1: Find the inverse of the function

Let $y = \log_3(x + 3)-7$. First, solve for $x$ in terms of $y$. Add 7 to both sides: $y + 7=\log_3(x + 3)$. By the definition of logarithms, if $y+7=\log_3(x + 3)$, then $3^{y + 7}=x + 3$. So, $x=3^{y + 7}-3$. The inverse function is $f^{-1}(x)=3^{x + 7}-3$.

Step2: Find lattice - points

Lattice points are points with integer coordinates. We can find some integer - valued points by choosing integer values of $x$ for the inverse function. When $x=-7$, $f^{-1}(-7)=3^{-7 + 7}-3=1 - 3=-2$. When $x=-6$, $f^{-1}(-6)=3^{-6 + 7}-3=3 - 3=0$. When $x=-5$, $f^{-1}(-5)=3^{-5 + 7}-3=9 - 3=6$.

Answer:

The lattice points of the inverse function are $(-7,-2),(-6,0),(-5,6)$ and other points can be found by substituting more integer values of $x$ into $f^{-1}(x)=3^{x + 7}-3$.