To solve the problem, we need to use the properties of even and odd functions.
### Even Function: [tex]\( f(x) \)[/tex]
An even function, [tex]\( f(x) \)[/tex], satisfies the condition:
[tex]\[ f(x) = f(-x) \][/tex]
Given the values for [tex]\( f \)[/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 use the property:
[tex]\[ f(2) = f(-2) \][/tex]
From the table, [tex]\( f(-2) = 4 \)[/tex]. Therefore:
[tex]\[ f(2) = 4 \][/tex]
Hence:
[tex]\[ a = 4 \][/tex]
### Odd Function: [tex]\( g(x) \)[/tex]
An odd function, [tex]\( g(x) \)[/tex], satisfies the condition:
[tex]\[ g(x) = -g(-x) \][/tex]
Given the values for [tex]\( g \)[/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 use the property:
[tex]\[ g(-2) = -g(2) \][/tex]
From the table, [tex]\( g(2) = -3 \)[/tex]. Therefore:
[tex]\[ g(-2) = -(-3) = 3 \][/tex]
Hence:
[tex]\[ b = 3 \][/tex]
### Summary of Results
- The value of [tex]\( a \)[/tex] is:
[tex]\[ a = 4 \][/tex]
- The value of [tex]\( b \)[/tex] is:
[tex]\[ b = 3 \][/tex]
Thus, the coordinates for the missing values have been determined as follows:
[tex]\[ a = 4 \][/tex]
[tex]\[ b = 3 \][/tex]
