Answer :
Alright, let's evaluate the function \( f(x) = 2x^3 - 3x^2 + 7 \) for the given values of \( x \).
### Step-by-Step Solution
#### Step 1: Evaluate \( f(-1) \)
To find \( f(-1) \):
1. Substitute \( x = -1 \) into the function.
[tex]\[ f(-1) = 2(-1)^3 - 3(-1)^2 + 7 \][/tex]
2. Calculate \( (-1)^3 \).
[tex]\[ (-1)^3 = -1 \][/tex]
3. Multiply by 2.
[tex]\[ 2(-1) = -2 \][/tex]
4. Calculate \( (-1)^2 \).
[tex]\[ (-1)^2 = 1 \][/tex]
5. Multiply by 3.
[tex]\[ 3(1) = 3 \][/tex]
6. Combine all the terms.
[tex]\[ f(-1) = -2 - 3 + 7 \][/tex]
7. Simplify.
[tex]\[ f(-1) = 2 \][/tex]
So, \( f(-1) = 2 \).
#### Step 2: Evaluate \( f(1) \)
To find \( f(1) \):
1. Substitute \( x = 1 \) into the function.
[tex]\[ f(1) = 2(1)^3 - 3(1)^2 + 7 \][/tex]
2. Calculate \( (1)^3 \).
[tex]\[ (1)^3 = 1 \][/tex]
3. Multiply by 2.
[tex]\[ 2(1) = 2 \][/tex]
4. Calculate \( (1)^2 \).
[tex]\[ (1)^2 = 1 \][/tex]
5. Multiply by 3.
[tex]\[ 3(1) = 3 \][/tex]
6. Combine all the terms.
[tex]\[ f(1) = 2 - 3 + 7 \][/tex]
7. Simplify.
[tex]\[ f(1) = 6 \][/tex]
So, \( f(1) = 6 \).
#### Step 3: Evaluate \( f(2) \)
To find \( f(2) \):
1. Substitute \( x = 2 \) into the function.
[tex]\[ f(2) = 2(2)^3 - 3(2)^2 + 7 \][/tex]
2. Calculate \( (2)^3 \).
[tex]\[ (2)^3 = 8 \][/tex]
3. Multiply by 2.
[tex]\[ 2(8) = 16 \][/tex]
4. Calculate \( (2)^2 \).
[tex]\[ (2)^2 = 4 \][/tex]
5. Multiply by 3.
[tex]\[ 3(4) = 12 \][/tex]
6. Combine all the terms.
[tex]\[ f(2) = 16 - 12 + 7 \][/tex]
7. Simplify.
[tex]\[ f(2) = 11 \][/tex]
So, \( f(2) = 11 \).
### Summary
The values of the function \( f(x) = 2x^3 - 3x^2 + 7 \) at the specified points are:
[tex]\[ \begin{array}{l} f(-1) = 2 \\ f(1) = 6 \\ f(2) = 11 \\ \end{array} \][/tex]
### Step-by-Step Solution
#### Step 1: Evaluate \( f(-1) \)
To find \( f(-1) \):
1. Substitute \( x = -1 \) into the function.
[tex]\[ f(-1) = 2(-1)^3 - 3(-1)^2 + 7 \][/tex]
2. Calculate \( (-1)^3 \).
[tex]\[ (-1)^3 = -1 \][/tex]
3. Multiply by 2.
[tex]\[ 2(-1) = -2 \][/tex]
4. Calculate \( (-1)^2 \).
[tex]\[ (-1)^2 = 1 \][/tex]
5. Multiply by 3.
[tex]\[ 3(1) = 3 \][/tex]
6. Combine all the terms.
[tex]\[ f(-1) = -2 - 3 + 7 \][/tex]
7. Simplify.
[tex]\[ f(-1) = 2 \][/tex]
So, \( f(-1) = 2 \).
#### Step 2: Evaluate \( f(1) \)
To find \( f(1) \):
1. Substitute \( x = 1 \) into the function.
[tex]\[ f(1) = 2(1)^3 - 3(1)^2 + 7 \][/tex]
2. Calculate \( (1)^3 \).
[tex]\[ (1)^3 = 1 \][/tex]
3. Multiply by 2.
[tex]\[ 2(1) = 2 \][/tex]
4. Calculate \( (1)^2 \).
[tex]\[ (1)^2 = 1 \][/tex]
5. Multiply by 3.
[tex]\[ 3(1) = 3 \][/tex]
6. Combine all the terms.
[tex]\[ f(1) = 2 - 3 + 7 \][/tex]
7. Simplify.
[tex]\[ f(1) = 6 \][/tex]
So, \( f(1) = 6 \).
#### Step 3: Evaluate \( f(2) \)
To find \( f(2) \):
1. Substitute \( x = 2 \) into the function.
[tex]\[ f(2) = 2(2)^3 - 3(2)^2 + 7 \][/tex]
2. Calculate \( (2)^3 \).
[tex]\[ (2)^3 = 8 \][/tex]
3. Multiply by 2.
[tex]\[ 2(8) = 16 \][/tex]
4. Calculate \( (2)^2 \).
[tex]\[ (2)^2 = 4 \][/tex]
5. Multiply by 3.
[tex]\[ 3(4) = 12 \][/tex]
6. Combine all the terms.
[tex]\[ f(2) = 16 - 12 + 7 \][/tex]
7. Simplify.
[tex]\[ f(2) = 11 \][/tex]
So, \( f(2) = 11 \).
### Summary
The values of the function \( f(x) = 2x^3 - 3x^2 + 7 \) at the specified points are:
[tex]\[ \begin{array}{l} f(-1) = 2 \\ f(1) = 6 \\ f(2) = 11 \\ \end{array} \][/tex]
