To determine if the function [tex]\( g \)[/tex] is invertible, we need to check if each value in the range of the function corresponds to exactly one value in the domain.
Looking at the given table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline$x$ & -3 & -2 & -1 & 0 & 1 \\
\hline$g(x)$ & 11 & 6 & 4 & 3 & 4 \\
\hline
\end{tabular}
\][/tex]
We can see that the function [tex]\( g \)[/tex] maps the following values:
[tex]\[
g(-3) = 11, \quad g(-2) = 6, \quad g(-1) = 4, \quad g(0) = 3, \quad g(1) = 4
\][/tex]
Notice that [tex]\( g(-1) = 4 \)[/tex] and [tex]\( g(1) = 4 \)[/tex]. Here, the value 4 in the range corresponds to two different values in the domain: [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]. This indicates that the function [tex]\( g \)[/tex] does not have a one-to-one mapping from the domain to the range.
For a function to be invertible, every element of the range should map back to exactly one unique element in the domain. However, in this case, the value 4 in the range is associated with two different values in the domain.
Thus, function [tex]\( g \)[/tex] does not appear to be invertible because it has repeating values in the range, specifically at [tex]\( g(-1) = g(1) = 4 \)[/tex]. This repeating value violates the one-to-one criterion necessary for a function to have an inverse.
