To evaluate [tex]\( f(6) \)[/tex] for the given function [tex]\( f(x) = \frac{2}{3}x^2 + \frac{1}{2}x - 2 \)[/tex], we follow these detailed steps:
1. Substitute [tex]\( x = 6 \)[/tex] into the function:
[tex]\[
f(6) = \frac{2}{3}(6)^2 + \frac{1}{2}(6) - 2
\][/tex]
2. Calculate the value inside the function:
- First, calculate [tex]\( (6)^2 \)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
- Next, substitute [tex]\( 36 \)[/tex] into the equation:
[tex]\[
\frac{2}{3} \cdot 36
\][/tex]
- Now, perform the multiplication:
[tex]\[
\frac{2 \cdot 36}{3} = \frac{72}{3} = 24
\][/tex]
3. Calculate the second term:
- Evaluate [tex]\( \frac{1}{2} \cdot 6 \)[/tex]:
[tex]\[
\frac{1 \cdot 6}{2} = \frac{6}{2} = 3
\][/tex]
4. Combine the results:
- Add the results from steps 2 and 3, and then subtract 2:
[tex]\[
f(6) = 24 + 3 - 2
\][/tex]
5. Perform the final arithmetic:
[tex]\[
24 + 3 = 27
\][/tex]
[tex]\[
27 - 2 = 25
\][/tex]
Thus, the value of [tex]\( f(6) \)[/tex] is [tex]\( 25 \)[/tex].
Therefore,
[tex]\[
f(6) = 25
\][/tex]
