Let's solve each problem step-by-step.
### Problem 1:
Given:
[tex]\[ f(x) = 3x + 7 \][/tex]
[tex]\[ g(x) = 2x - 1 \][/tex]
To find the sum [tex]\( (F+g)(x) \)[/tex]:
[tex]\[ (F+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ = (3x + 7) + (2x - 1) \][/tex]
[tex]\[ = 3x + 2x + 7 - 1 \][/tex]
[tex]\[ = 5x + 6 \][/tex]
To find the product [tex]\( (f \cdot g)(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ = (3x + 7) \cdot (2x - 1) \][/tex]
[tex]\[ = 3x \cdot 2x + 3x \cdot (-1) + 7 \cdot 2x + 7 \cdot (-1) \][/tex]
[tex]\[ = 6x^2 - 3x + 14x - 7 \][/tex]
[tex]\[ = 6x^2 + 11x - 7 \][/tex]
For [tex]\( x = 1 \)[/tex]:
[tex]\[ (F+g)(1) = 5 \cdot 1 + 6 = 11 \][/tex]
[tex]\[ (f \cdot g)(1) = 6 \cdot 1^2 + 11 \cdot 1 - 7 = 6 + 11 - 7 = 10 \][/tex]
### Problem 2:
Given:
[tex]\[ f(x) = 2x^2 - 3x + 5 \][/tex]
[tex]\[ g(x) = x^2 - 2x + 2 \][/tex]
To find the sum [tex]\( (F+g)(x) \)[/tex]:
[tex]\[ (F+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ = (2x^2 - 3x + 5) + (x^2 - 2x + 2) \][/tex]
[tex]\[ = 2x^2 + x^2 - 3x - 2x + 5 + 2 \][/tex]
[tex]\[ = 3x^2 - 5x + 7 \][/tex]
To find the product [tex]\( (f \cdot g)(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ = (2x^2 - 3x + 5) \cdot (x^2 - 2x + 2) \][/tex]
[tex]\[ = 2x^2 \cdot x^2 + 2x^2 \cdot (-2x) + 2x^2 \cdot 2 + (-3x) \cdot x^2 + (-3x) \cdot (-2x) + (-3x) \cdot 2 + 5 \cdot x^2 + 5 \cdot (-2x) + 5 \cdot 2 \][/tex]
Hence, the product is a lengthy polynomial expression.
For [tex]\( x = 1 \)[/tex]:
[tex]\[ (F+g)(1) = 3 \cdot 1^2 - 5 \cdot 1 + 7 = 3 - 5 + 7 = 5 \][/tex]
[tex]\[ (f \cdot g)(1) = 4 \][/tex]
### Problem 3:
Given:
[tex]\[ f(x) = x^3 - 3x^2 - x + 1 \][/tex]
[tex]\[ g(x) = x^3 + 2x^2 + 2x + 8 \][/tex]
To find the sum [tex]\( (F+g)(x) \)[/tex]:
[tex]\[ (F+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ = (x^3 - 3x^2 - x + 1) + (x^3 + 2x^2 + 2x + 8) \][/tex]
[tex]\[ = x^3 + x^3 - 3x^2 + 2x^2 - x + 2x + 1 + 8 \][/tex]
[tex]\[ = 2x^3 - x^2 + x + 9 \][/tex]
To find the product [tex]\( (f \cdot g)(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ = (x^3 - 3x^2 - x + 1) \cdot (x^3 + 2x^2 + 2x + 8) \][/tex]
This is also a lengthy expansion of polynomial multiplication.
For [tex]\( x = 1 \)[/tex]:
[tex]\[ (F+g)(1) = 2 \cdot 1^3 - 1^2 + 1 + 9 = 2 - 1 + 1 + 9 = 11 \][/tex]
[tex]\[ (f \cdot g)(1) = -26 \][/tex]
Summarized results:
1. [tex]\( (F+g)(1) = 11\)[/tex], [tex]\( (f \cdot g)(1) = 10 \)[/tex]
2. [tex]\( (F+g)(1) = 5 \)[/tex], [tex]\( (f \cdot g)(1) = 4 \)[/tex]
3. [tex]\( (F+g)(1) = 11 \)[/tex], [tex]\( (f \cdot g)(1) = -26 \)[/tex]
