To find the solution to the system of equations for [tex]\( f(x) = 2x^2 + x + 4 \)[/tex] and [tex]\( g(x) \)[/tex], where the values of [tex]\( g(x) \)[/tex] are given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-2 & 1 \\
-1 & 3 \\
0 & 5 \\
1 & 7 \\
2 & 9 \\
\hline
\end{array}
\][/tex]
we need to determine where the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are equal. We will check each value of [tex]\( x \)[/tex] given in the table.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 2(-2)^2 + (-2) + 4 = 2(4) - 2 + 4 = 8 - 2 + 4 = 10
\][/tex]
Therefore, [tex]\( f(-2) = 10 \)[/tex] and [tex]\( g(-2) = 1 \)[/tex]. They are not equal.
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2(-1)^2 + (-1) + 4 = 2(1) - 1 + 4 = 2 - 1 + 4 = 5
\][/tex]
Therefore, [tex]\( f(-1) = 5 \)[/tex] and [tex]\( g(-1) = 3 \)[/tex]. They are not equal.
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2(0)^2 + 0 + 4 = 0 + 0 + 4 = 4
\][/tex]
Therefore, [tex]\( f(0) = 4 \)[/tex] and [tex]\( g(0) = 5 \)[/tex]. They are not equal.
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2(1)^2 + 1 + 4 = 2(1) + 1 + 4 = 2 + 1 + 4 = 7
\][/tex]
Therefore, [tex]\( f(1) = 7 \)[/tex] and [tex]\( g(1) = 7 \)[/tex]. They are equal.
Since [tex]\( f(1) = g(1) \)[/tex], we find that [tex]\( x = 1 \)[/tex] is a solution to the system of equations. This gives us the point [tex]\( (1, 7) \)[/tex] which satisfies both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2(2)^2 + 2 + 4 = 2(4) + 2 + 4 = 8 + 2 + 4 = 14
\][/tex]
Therefore, [tex]\( f(2) = 14 \)[/tex] and [tex]\( g(2) = 9 \)[/tex]. They are not equal.
Thus, the solution to the system of equations is:
[tex]\[
\boxed{(1, 7)}
\][/tex]
