To compare the maximum values of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], let's analyze each function step by step.
### Step 1: Analyze [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -2x^2 + 9 \)[/tex] is a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex]. Since the coefficient of [tex]\( x^2 \)[/tex] is negative [tex]\((-2)\)[/tex], the parabola opens downwards, indicating that the vertex represents the maximum value of the function.
The vertex form of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is found at [tex]\( x = -\frac{b}{2a} \)[/tex]. For [tex]\( f(x) \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 0 \)[/tex]
- [tex]\( c = 9 \)[/tex]
The vertex lies at:
[tex]\[ x = -\frac{0}{2(-2)} = 0 \][/tex]
To find the maximum value of [tex]\( f(x) \)[/tex], substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = -2(0)^2 + 9 = 9 \][/tex]
Thus, the maximum value of [tex]\( f(x) \)[/tex] is 9.
### Step 2: Analyze [tex]\( g(x) \)[/tex]
The function [tex]\( g(x) \)[/tex] is given by a table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
0 & -5 \\
\hline
1 & 3 \\
\hline
2 & 11 \\
\hline
3 & 3 \\
\hline
4 & -5 \\
\hline
\end{array}
\][/tex]
To find the maximum value of [tex]\( g(x) \)[/tex], we simply look for the highest value in the given entries. From the table:
- [tex]\( g(0) = -5 \)[/tex]
- [tex]\( g(1) = 3 \)[/tex]
- [tex]\( g(2) = 11 \)[/tex]
- [tex]\( g(3) = 3 \)[/tex]
- [tex]\( g(4) = -5 \)[/tex]
The maximum value of [tex]\( g(x) \)[/tex] is 11.
### Step 3: Compare the Maximum Values
- The maximum value of [tex]\( f(x) \)[/tex] is 9.
- The maximum value of [tex]\( g(x) \)[/tex] is 11.
By comparing these values, we see that the maximum value of [tex]\( g(x) \)[/tex] is greater than the maximum value of [tex]\( f(x) \)[/tex].
### Conclusion
The statement that best compares the maximum values of the two functions is:
[tex]\[ \boxed{\$g(x)\$ \text{ has a greater maximum value than } \$f(x)\$.} \][/tex]
