Answer :
To complete the exercises using the function [tex]\( f(x) = 2x^3 - 3x^2 + 7 \)[/tex], we will substitute the given values of [tex]\( x \)[/tex] into the function and simplify according to algebraic operations.
### Step-by-Step Solutions:
1. Calculate [tex]\( f(-1) \)[/tex]:
[tex]\[ \begin{aligned} f(-1) = & \, 2(-1)^3 - 3(-1)^2 + 7 \\ = & \, 2(-1) - 3(1) + 7 \\ = & \, -2 - 3 + 7 \\ = & \, 2. \end{aligned} \][/tex]
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ \begin{aligned} f(1) = & \, 2(1)^3 - 3(1)^2 + 7 \\ = & \, 2(1) - 3(1) + 7 \\ = & \, 2 - 3 + 7 \\ = & \, 6. \end{aligned} \][/tex]
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ \begin{aligned} f(2) = & \, 2(2)^3 - 3(2)^2 + 7 \\ = & \, 2(8) - 3(4) + 7 \\ = & \, 16 - 12 + 7 \\ = & \, 11. \end{aligned} \][/tex]
### Summary of Final Answers:
[tex]\[ \begin{aligned} f(-1) &= 2, \\ f(1) &= 6, \\ f(2) &= 11. \end{aligned} \][/tex]
### Step-by-Step Solutions:
1. Calculate [tex]\( f(-1) \)[/tex]:
[tex]\[ \begin{aligned} f(-1) = & \, 2(-1)^3 - 3(-1)^2 + 7 \\ = & \, 2(-1) - 3(1) + 7 \\ = & \, -2 - 3 + 7 \\ = & \, 2. \end{aligned} \][/tex]
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ \begin{aligned} f(1) = & \, 2(1)^3 - 3(1)^2 + 7 \\ = & \, 2(1) - 3(1) + 7 \\ = & \, 2 - 3 + 7 \\ = & \, 6. \end{aligned} \][/tex]
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ \begin{aligned} f(2) = & \, 2(2)^3 - 3(2)^2 + 7 \\ = & \, 2(8) - 3(4) + 7 \\ = & \, 16 - 12 + 7 \\ = & \, 11. \end{aligned} \][/tex]
### Summary of Final Answers:
[tex]\[ \begin{aligned} f(-1) &= 2, \\ f(1) &= 6, \\ f(2) &= 11. \end{aligned} \][/tex]
