Answer :
Sure! Let's go through each pair of functions step-by-step to find the sum [tex]\( (f+g)(x) \)[/tex] and the product [tex]\( (f \cdot g)(x) \)[/tex].
### Pair 1: [tex]\( f(x) = 3x + 7 \)[/tex] and [tex]\( g(x) = 2x - 1 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (3x + 7) + (2x - 1) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = 3x + 2x + 7 - 1 \][/tex]
[tex]\[ (f+g)(x) = 5x + 6 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions:
[tex]\[ f(x) \cdot g(x) = (3x + 7) \cdot (2x - 1) \][/tex]
Using the distributive property:
[tex]\[ f(x) \cdot g(x) = 3x \cdot 2x + 3x \cdot (-1) + 7 \cdot 2x + 7 \cdot (-1) \][/tex]
[tex]\[ f(x) \cdot g(x) = 6x^2 - 3x + 14x - 7 \][/tex]
Combining like terms:
[tex]\[ (f \cdot g)(x) = 6x^2 + 11x - 7 \][/tex]
### Pair 2: [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex] and [tex]\( g(x) = x^2 - 2x + 2 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (2x^2 - 3x + 5) + (x^2 - 2x + 2) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = 2x^2 + x^2 - 3x - 2x + 5 + 2 \][/tex]
[tex]\[ (f+g)(x) = 3x^2 - 5x + 7 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions:
[tex]\[ f(x) \cdot g(x) = (2x^2 - 3x + 5) \cdot (x^2 - 2x + 2) \][/tex]
Expanding the product requires distributing each term:
[tex]\[ f(x) \cdot g(x) = 2x^2 \cdot (x^2 - 2x + 2) - 3x \cdot (x^2 - 2x + 2) + 5 \cdot (x^2 - 2x + 2) \][/tex]
[tex]\[ f(x) \cdot g(x) = 2x^4 - 4x^3 + 4x^2 - 3x^3 + 6x^2 - 6x + 5x^2 - 10x + 10 \][/tex]
Combining like terms:
[tex]\[ (f \cdot g)(x) = 2x^4 - 7x^3 + 15x^2 - 16x + 10 \][/tex]
### Pair 3: [tex]\( f(x) = x^3 - 3x^2 - x + 1 \)[/tex] and [tex]\( g(x) = x^3 + 2x^2 + 2x + 8 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (x^3 - 3x^2 - x + 1) + (x^3 + 2x^2 + 2x + 8) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = x^3 + x^3 - 3x^2 + 2x^2 - x + 2x + 1 + 8 \][/tex]
[tex]\[ (f+g)(x) = 2x^3 - x^2 + x + 9 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions using the distributive property:
[tex]\[ f(x) \cdot g(x) = (x^3 - 3x^2 - x + 1) \cdot (x^3 + 2x^2 + 2x + 8) \][/tex]
Expanding the product term by term:
[tex]\[ f(x) \cdot g(x) = x^3 \cdot (x^3 + 2x^2 + 2x + 8) - 3x^2 \cdot (x^3 + 2x^2 + 2x + 8) - x \cdot (x^3 + 2x^2 + 2x + 8) + 1 \cdot (x^3 + 2x^2 + 2x + 8) \][/tex]
[tex]\[ = x^6 + 2x^5 + 2x^4 + 8x^3 - 3x^5 - 6x^4 - 6x^3 - 24x^2 - x^4 - 2x^3 - 2x^2 - 8x + x^3 + 2x^2 + 2x + 8 \][/tex]
Combining like terms:
[tex]\[ f(x) \cdot g(x) = x^6 - x^5 - 5x^4 + x^3 - 24x^2 - 6x + 8 \][/tex]
These sums and products give us the functions for each pair of functions.
### Pair 1: [tex]\( f(x) = 3x + 7 \)[/tex] and [tex]\( g(x) = 2x - 1 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (3x + 7) + (2x - 1) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = 3x + 2x + 7 - 1 \][/tex]
[tex]\[ (f+g)(x) = 5x + 6 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions:
[tex]\[ f(x) \cdot g(x) = (3x + 7) \cdot (2x - 1) \][/tex]
Using the distributive property:
[tex]\[ f(x) \cdot g(x) = 3x \cdot 2x + 3x \cdot (-1) + 7 \cdot 2x + 7 \cdot (-1) \][/tex]
[tex]\[ f(x) \cdot g(x) = 6x^2 - 3x + 14x - 7 \][/tex]
Combining like terms:
[tex]\[ (f \cdot g)(x) = 6x^2 + 11x - 7 \][/tex]
### Pair 2: [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex] and [tex]\( g(x) = x^2 - 2x + 2 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (2x^2 - 3x + 5) + (x^2 - 2x + 2) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = 2x^2 + x^2 - 3x - 2x + 5 + 2 \][/tex]
[tex]\[ (f+g)(x) = 3x^2 - 5x + 7 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions:
[tex]\[ f(x) \cdot g(x) = (2x^2 - 3x + 5) \cdot (x^2 - 2x + 2) \][/tex]
Expanding the product requires distributing each term:
[tex]\[ f(x) \cdot g(x) = 2x^2 \cdot (x^2 - 2x + 2) - 3x \cdot (x^2 - 2x + 2) + 5 \cdot (x^2 - 2x + 2) \][/tex]
[tex]\[ f(x) \cdot g(x) = 2x^4 - 4x^3 + 4x^2 - 3x^3 + 6x^2 - 6x + 5x^2 - 10x + 10 \][/tex]
Combining like terms:
[tex]\[ (f \cdot g)(x) = 2x^4 - 7x^3 + 15x^2 - 16x + 10 \][/tex]
### Pair 3: [tex]\( f(x) = x^3 - 3x^2 - x + 1 \)[/tex] and [tex]\( g(x) = x^3 + 2x^2 + 2x + 8 \)[/tex]
Sum: [tex]\( (f+g)(x) \)[/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the two functions:
[tex]\[ f(x) + g(x) = (x^3 - 3x^2 - x + 1) + (x^3 + 2x^2 + 2x + 8) \][/tex]
Combining like terms:
[tex]\[ (f+g)(x) = x^3 + x^3 - 3x^2 + 2x^2 - x + 2x + 1 + 8 \][/tex]
[tex]\[ (f+g)(x) = 2x^3 - x^2 + x + 9 \][/tex]
Product: [tex]\( (f \cdot g)(x) \)[/tex]
To find [tex]\( (f \cdot g)(x) \)[/tex], we multiply the two functions using the distributive property:
[tex]\[ f(x) \cdot g(x) = (x^3 - 3x^2 - x + 1) \cdot (x^3 + 2x^2 + 2x + 8) \][/tex]
Expanding the product term by term:
[tex]\[ f(x) \cdot g(x) = x^3 \cdot (x^3 + 2x^2 + 2x + 8) - 3x^2 \cdot (x^3 + 2x^2 + 2x + 8) - x \cdot (x^3 + 2x^2 + 2x + 8) + 1 \cdot (x^3 + 2x^2 + 2x + 8) \][/tex]
[tex]\[ = x^6 + 2x^5 + 2x^4 + 8x^3 - 3x^5 - 6x^4 - 6x^3 - 24x^2 - x^4 - 2x^3 - 2x^2 - 8x + x^3 + 2x^2 + 2x + 8 \][/tex]
Combining like terms:
[tex]\[ f(x) \cdot g(x) = x^6 - x^5 - 5x^4 + x^3 - 24x^2 - 6x + 8 \][/tex]
These sums and products give us the functions for each pair of functions.
