Answer :
Sure, let's go through the calculations step-by-step for each function and the provided results.
### Given Functions:
1. [tex]\( f(x) = 3x - 1 \)[/tex]
2. [tex]\( g(x) = 2x + 3 \)[/tex]
3. [tex]\( v(x) = x \times 7 \)[/tex]
4. [tex]\( p(x) = x + 10 \)[/tex]
### Perform Calculations:
1. Sum of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f+g \)[/tex]:
[tex]\[ f(x) + g(x) = (3x - 1) + (2x + 3) = 3x - 1 + 2x + 3 = 5x + 2 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 5(1) + 2 = 5 + 2 = 7 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 5(2) + 2 = 10 + 2 = 12 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 5(3) + 2 = 15 + 2 = 17 \)[/tex]
So, [tex]\( f+g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([7, 12, 17]\)[/tex].
2. Difference of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f-g \)[/tex]:
[tex]\[ f(x) - g(x) = (3x - 1) - (2x + 3) = 3x - 1 - 2x - 3 = x - 4 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 - 4 = -3 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 - 4 = -2 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 - 4 = -1 \)[/tex]
So, [tex]\( f-g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([-3, -2, -1]\)[/tex].
3. Quotient of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f/g \)[/tex]:
[tex]\[ \frac{f(x)}{g(x)} = \frac{3x - 1}{2x + 3} \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( \frac{3(1) - 1}{2(1) + 3} = \frac{2}{5} = 0.4 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( \frac{3(2) - 1}{2(2) + 3} = \frac{5}{7} \approx 0.714 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( \frac{3(3) - 1}{2(3) + 3} = \frac{8}{9} \approx 0.889 \)[/tex]
So, [tex]\( f/g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([0.4, 0.714, 0.889]\)[/tex].
4. Evaluation of [tex]\( v(x) \)[/tex]:
[tex]\[ v(x) = x \times 7 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 \times 7 = 7 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 \times 7 = 14 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 \times 7 = 21 \)[/tex]
So, [tex]\( v(x) \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([7, 14, 21]\)[/tex].
5. Evaluation of [tex]\( p(x) \)[/tex]:
[tex]\[ p(x) = x + 10 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 + 10 = 11 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 + 10 = 12 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 + 10 = 13 \)[/tex]
So, [tex]\( p(x) \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([11, 12, 13]\)[/tex].
### Summary of Results:
- [tex]\( f+g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([7, 12, 17]\)[/tex]
- [tex]\( f-g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([-3, -2, -1]\)[/tex]
- [tex]\( f/g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([0.4, 0.714, 0.889]\)[/tex]
- [tex]\( v(x) \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([7, 14, 21]\)[/tex]
- [tex]\( p(x) \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([11, 12, 13]\)[/tex]
So, the detailed calculations produce the expected results.
### Given Functions:
1. [tex]\( f(x) = 3x - 1 \)[/tex]
2. [tex]\( g(x) = 2x + 3 \)[/tex]
3. [tex]\( v(x) = x \times 7 \)[/tex]
4. [tex]\( p(x) = x + 10 \)[/tex]
### Perform Calculations:
1. Sum of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f+g \)[/tex]:
[tex]\[ f(x) + g(x) = (3x - 1) + (2x + 3) = 3x - 1 + 2x + 3 = 5x + 2 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 5(1) + 2 = 5 + 2 = 7 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 5(2) + 2 = 10 + 2 = 12 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 5(3) + 2 = 15 + 2 = 17 \)[/tex]
So, [tex]\( f+g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([7, 12, 17]\)[/tex].
2. Difference of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f-g \)[/tex]:
[tex]\[ f(x) - g(x) = (3x - 1) - (2x + 3) = 3x - 1 - 2x - 3 = x - 4 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 - 4 = -3 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 - 4 = -2 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 - 4 = -1 \)[/tex]
So, [tex]\( f-g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([-3, -2, -1]\)[/tex].
3. Quotient of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], denoted as [tex]\( f/g \)[/tex]:
[tex]\[ \frac{f(x)}{g(x)} = \frac{3x - 1}{2x + 3} \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( \frac{3(1) - 1}{2(1) + 3} = \frac{2}{5} = 0.4 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( \frac{3(2) - 1}{2(2) + 3} = \frac{5}{7} \approx 0.714 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( \frac{3(3) - 1}{2(3) + 3} = \frac{8}{9} \approx 0.889 \)[/tex]
So, [tex]\( f/g \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([0.4, 0.714, 0.889]\)[/tex].
4. Evaluation of [tex]\( v(x) \)[/tex]:
[tex]\[ v(x) = x \times 7 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 \times 7 = 7 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 \times 7 = 14 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 \times 7 = 21 \)[/tex]
So, [tex]\( v(x) \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([7, 14, 21]\)[/tex].
5. Evaluation of [tex]\( p(x) \)[/tex]:
[tex]\[ p(x) = x + 10 \][/tex]
For the values [tex]\( x = 1, 2, 3 \)[/tex]:
- When [tex]\( x = 1 \)[/tex]: [tex]\( 1 + 10 = 11 \)[/tex]
- When [tex]\( x = 2 \)[/tex]: [tex]\( 2 + 10 = 12 \)[/tex]
- When [tex]\( x = 3 \)[/tex]: [tex]\( 3 + 10 = 13 \)[/tex]
So, [tex]\( p(x) \)[/tex] for [tex]\( x = 1, 2, 3 \)[/tex] is [tex]\([11, 12, 13]\)[/tex].
### Summary of Results:
- [tex]\( f+g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([7, 12, 17]\)[/tex]
- [tex]\( f-g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([-3, -2, -1]\)[/tex]
- [tex]\( f/g \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([0.4, 0.714, 0.889]\)[/tex]
- [tex]\( v(x) \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([7, 14, 21]\)[/tex]
- [tex]\( p(x) \)[/tex] at [tex]\( x = [1, 2, 3] \)[/tex] gives [tex]\([11, 12, 13]\)[/tex]
So, the detailed calculations produce the expected results.
