To determine after which value of \( x \) the function \( g \) exceeds function \( f \), let's compare the values of \( f(x) \) and \( g(x) \) at each \( x \) given in the table.
We will look at the values of \( f(x) \) and \( g(x) \) side by side and find the first instance where \( g(x) \) is greater than \( f(x) \):
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
-1 & -2.5 & -3.67 \\
\hline
0 & -1 & -3 \\
\hline
1 & 1.5 & -1 \\
\hline
2 & 5 & 5 \\
\hline
3 & 9.5 & 23 \\
\hline
4 & 15 & 77 \\
\hline
5 & 21.5 & 239 \\
\hline
\end{array}
\][/tex]
1. For \( x = -1 \):
[tex]\[
f(-1) = -2.5, \quad g(-1) = -3.67
\][/tex]
\( g(-1) \) is not greater than \( f(-1) \).
2. For \( x = 0 \):
[tex]\[
f(0) = -1, \quad g(0) = -3
\][/tex]
\( g(0) \) is not greater than \( f(0) \).
3. For \( x = 1 \):
[tex]\[
f(1) = 1.5, \quad g(1) = -1
\][/tex]
\( g(1) \) is not greater than \( f(1) \).
4. For \( x = 2 \):
[tex]\[
f(2) = 5, \quad g(2) = 5
\][/tex]
\( g(2) \) is equal to \( f(2) \).
5. For \( x = 3 \):
[tex]\[
f(3) = 9.5, \quad g(3) = 23
\][/tex]
\( g(3) \) is greater than \( f(3) \).
At \( x = 3 \), for the first time, \( g(x) \) exceeds \( f(x) \).
Thus, function [tex]\( g \)[/tex] exceeds function [tex]\( f \)[/tex] at [tex]\( x = 3 \)[/tex] and for all subsequent [tex]\( x \)[/tex] values.
