To solve the question, we need to evaluate the functions [tex]\( f(x) = -\frac{2}{3} x + 3 \)[/tex] and [tex]\( g(x) = 2 x^2 - 5 \)[/tex] for the given values of [tex]\( x \)[/tex]. Here are the detailed steps:
1. For [tex]\( f(x) \)[/tex]:
- Calculate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = -\frac{2}{3} \cdot 3 + 3
\][/tex]
The value of [tex]\( f(3) \)[/tex] is 1.
- Calculate [tex]\( f(-5) \)[/tex]:
[tex]\[
f(-5) = -\frac{2}{3} \cdot (-5) + 3
\][/tex]
The value of [tex]\( f(-5) \)[/tex] is approximately 6.333333333333333 which is close to 6.33 (infinite repeating decimal).
2. For [tex]\( g(x) \)[/tex]:
- Calculate [tex]\( g(-3) \)[/tex]:
[tex]\[
g(-3) = 2 \cdot (-3)^2 - 5
\][/tex]
The value of [tex]\( g(-3) \)[/tex] is 13.
- Calculate [tex]\( g(-7) \)[/tex]:
[tex]\[
g(-7) = 2 \cdot (-7)^2 - 5
\][/tex]
The value of [tex]\( g(-7) \)[/tex] is 93.
- Calculate [tex]\( g(8) \)[/tex]:
[tex]\[
g(8) = 2 \cdot 8^2 - 5
\][/tex]
The value of [tex]\( g(8) \)[/tex] is 123.
- Calculate [tex]\( g(9) \)[/tex]:
[tex]\[
g(9) = 2 \cdot 9^2 - 5
\][/tex]
The value of [tex]\( g(9) \)[/tex] is 157.
3. Fill in the table with our calculated values:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
f(3) & 1 & - \\
\hline
f(-5) & 6.33 & - \\
\hline
g(-3) & - & 13 \\
\hline
g(-7) & - & 93 \\
\hline
g(8) & - & 123 \\
\hline
g(9) & - & 157 \\
\hline
\end{array}
\][/tex]
So, the table should look like this:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
f(3) & 1 & - \\
\hline
f(-5) & 6.33 & - \\
\hline
g(-3) & - & 13 \\
\hline
g(-7) & - & 93 \\
\hline
g(8) & - & 123 \\
\hline
g(9) & - & 157 \\
\hline
\end{array}
\][/tex]
The correct values should be placed accordingly to ensure clarity in the table.
