Let's first solve for [tex]\(a\)[/tex] for the function [tex]\(f(x)\)[/tex].
We know that [tex]\(f\)[/tex] is an even function. By definition, even functions satisfy the following property:
[tex]\[ f(-x) = f(x) \][/tex]
Given the table for [tex]\(f(x)\)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & 4 \\
\hline
0 & 5 \\
\hline
2 & $a$ \\
\hline
3 & 7 \\
\hline
\end{tabular}
\][/tex]
To find [tex]\(a\)[/tex], we look at the entry where [tex]\(x = -2\)[/tex] and [tex]\(f(x) = 4\)[/tex]. Since [tex]\(f\)[/tex] is even:
[tex]\[ f(-2) = f(2) \][/tex]
Hence:
[tex]\[ f(2) = 4 \][/tex]
Thus,
[tex]\[ a = 4 \][/tex]
So, the value of [tex]\(a\)[/tex] is 4.
Now let's solve for [tex]\(b\)[/tex] for the function [tex]\(g(x)\)[/tex].
We know that [tex]\(g\)[/tex] is an odd function. By definition, odd functions satisfy the following property:
[tex]\[ g(-x) = -g(x) \][/tex]
Given the table for [tex]\(g(x)\)[/tex]:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $g(x)$ \\
\hline
-2 & $b$ \\
\hline
0 & 0 \\
\hline
2 & -3 \\
\hline
3 & -4 \\
\hline
\end{tabular}
\][/tex]
To find [tex]\(b\)[/tex], we look at the entry where [tex]\(x = 2\)[/tex] and [tex]\(g(x) = -3\)[/tex]. Since [tex]\(g\)[/tex] is odd:
[tex]\[ g(-2) = -g(2) \][/tex]
Hence:
[tex]\[ g(-2) = -(-3) = 3 \][/tex]
Thus,
[tex]\[ b = 3 \][/tex]
So, the value of [tex]\(b\)[/tex] is 3.
In conclusion:
[tex]\[ a = 4 \][/tex]
[tex]\[ b = 3 \][/tex]
