To evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 6 \)[/tex], we will plug in the value of [tex]\( x = 6 \)[/tex] into the function [tex]\( f(x) = \frac{2}{3} x^2 + \frac{1}{2} x - 2 \)[/tex] and simplify.
Here are the steps to find [tex]\( f(6) \)[/tex]:
1. Start with the function [tex]\( f(x) = \frac{2}{3} x^2 + \frac{1}{2} x - 2 \)[/tex].
2. Substitute [tex]\( x = 6 \)[/tex] into the function:
[tex]\[
f(6) = \frac{2}{3} (6)^2 + \frac{1}{2} (6) - 2
\][/tex]
3. Calculate [tex]\( (6)^2 \)[/tex]:
[tex]\[
(6)^2 = 36
\][/tex]
4. Now substitute the value back into the expression:
[tex]\[
f(6) = \frac{2}{3} \cdot 36 + \frac{1}{2} \cdot 6 - 2
\][/tex]
5. Multiply [tex]\(\frac{2}{3}\)[/tex] by 36:
[tex]\[
\frac{2}{3} \cdot 36 = 24
\][/tex]
6. Multiply [tex]\(\frac{1}{2}\)[/tex] by 6:
[tex]\[
\frac{1}{2} \cdot 6 = 3
\][/tex]
7. Now sum up the terms:
[tex]\[
f(6) = 24 + 3 - 2
\][/tex]
8. Combine the terms:
[tex]\[
24 + 3 = 27
\][/tex]
9. Subtract 2 from 27:
[tex]\[
27 - 2 = 25
\][/tex]
Therefore, the value of [tex]\( f(6) \)[/tex] is:
[tex]\[
f(6) = 25
\][/tex]
