Answer :
Certainly! Let's go through each part of the question step-by-step:
### i. [tex]\((s \cdot g)(1)\)[/tex]
To find [tex]\((s \cdot g)(1)\)[/tex], we need to evaluate the product of the functions [tex]\(s\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 1\)[/tex]:
1. Evaluate [tex]\(s(1)\)[/tex].
2. Evaluate [tex]\(g(1)\)[/tex].
3. Multiply the results from steps 1 and 2.
From the computation:
- [tex]\( s(1) = 2(1) + 1 = 3 \)[/tex]
- [tex]\( g(1) = 1^2 - 3 = -2 \)[/tex]
So, the product is:
[tex]\[ (s \cdot g)(1) = s(1) \cdot g(1) = 3 \cdot (-2) = -6 \][/tex]
Hence, [tex]\((s \cdot g)(1) = -6\)[/tex].
### ii. [tex]\((g - s)(3)\)[/tex]
To find [tex]\((g - s)(3)\)[/tex], we need to evaluate the difference of the functions [tex]\(g\)[/tex] and [tex]\(s\)[/tex] at [tex]\(x = 3\)[/tex]:
1. Evaluate [tex]\(g(3)\)[/tex].
2. Evaluate [tex]\(s(3)\)[/tex].
3. Subtract the result of [tex]\(s(3)\)[/tex] from [tex]\(g(3)\)[/tex].
From the computation:
- [tex]\( g(3) = 3^2 - 3 = 9 - 3 = 6 \)[/tex]
- [tex]\( s(3) = 2(3) + 1 = 6 + 1 = 7 \)[/tex]
So, the difference is:
[tex]\[ (g - s)(3) = g(3) - s(3) = 6 - 7 = -1 \][/tex]
Hence, [tex]\((g - s)(3) = -1\)[/tex].
### iii. [tex]\((s \circ g)(1.5)\)[/tex]
To find [tex]\((s \circ g)(1.5)\)[/tex], we need to compute the composition of [tex]\(s\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 1.5\)[/tex], which means evaluating [tex]\(s\)[/tex] at [tex]\(g(1.5)\)[/tex]:
1. Evaluate [tex]\(g(1.5)\)[/tex].
2. Then, evaluate [tex]\(s\)[/tex] at the result from step 1.
From the computation:
- [tex]\( g(1.5) = (1.5)^2 - 3 = 2.25 - 3 = -0.75 \)[/tex]
- [tex]\( s(-0.75) = 2(-0.75) + 1 = -1.5 + 1 = -0.5 \)[/tex]
So, the composition is:
[tex]\[ (s \circ g)(1.5) = s(g(1.5)) = s(-0.75) = -0.5 \][/tex]
Hence, [tex]\((s \circ g)(1.5) = -0.5\)[/tex].
### iv. [tex]\((g \circ s)(-4)\)[/tex]
To find [tex]\((g \circ s)(-4)\)[/tex], we need to compute the composition of [tex]\(g\)[/tex] and [tex]\(s\)[/tex] at [tex]\(x = -4\)[/tex], which means evaluating [tex]\(g\)[/tex] at [tex]\(s(-4)\)[/tex]:
1. Evaluate [tex]\(s(-4)\)[/tex].
2. Then, evaluate [tex]\(g\)[/tex] at the result from step 1.
From the computation:
- [tex]\( s(-4) = 2(-4) + 1 = -8 + 1 = -7 \)[/tex]
- [tex]\( g(-7) = (-7)^2 - 3 = 49 - 3 = 46 \)[/tex]
So, the composition is:
[tex]\[ (g \circ s)(-4) = g(s(-4)) = g(-7) = 46 \][/tex]
Hence, [tex]\((g \circ s)(-4) = 46\)[/tex].
In summary:
i. [tex]\((s \cdot g)(1) = -6\)[/tex]
ii. [tex]\((g - s)(3) = -1\)[/tex]
iii. [tex]\((s \circ g)(1.5) = -0.5\)[/tex]
iv. [tex]\((g \circ s)(-4) = 46\)[/tex]
### i. [tex]\((s \cdot g)(1)\)[/tex]
To find [tex]\((s \cdot g)(1)\)[/tex], we need to evaluate the product of the functions [tex]\(s\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 1\)[/tex]:
1. Evaluate [tex]\(s(1)\)[/tex].
2. Evaluate [tex]\(g(1)\)[/tex].
3. Multiply the results from steps 1 and 2.
From the computation:
- [tex]\( s(1) = 2(1) + 1 = 3 \)[/tex]
- [tex]\( g(1) = 1^2 - 3 = -2 \)[/tex]
So, the product is:
[tex]\[ (s \cdot g)(1) = s(1) \cdot g(1) = 3 \cdot (-2) = -6 \][/tex]
Hence, [tex]\((s \cdot g)(1) = -6\)[/tex].
### ii. [tex]\((g - s)(3)\)[/tex]
To find [tex]\((g - s)(3)\)[/tex], we need to evaluate the difference of the functions [tex]\(g\)[/tex] and [tex]\(s\)[/tex] at [tex]\(x = 3\)[/tex]:
1. Evaluate [tex]\(g(3)\)[/tex].
2. Evaluate [tex]\(s(3)\)[/tex].
3. Subtract the result of [tex]\(s(3)\)[/tex] from [tex]\(g(3)\)[/tex].
From the computation:
- [tex]\( g(3) = 3^2 - 3 = 9 - 3 = 6 \)[/tex]
- [tex]\( s(3) = 2(3) + 1 = 6 + 1 = 7 \)[/tex]
So, the difference is:
[tex]\[ (g - s)(3) = g(3) - s(3) = 6 - 7 = -1 \][/tex]
Hence, [tex]\((g - s)(3) = -1\)[/tex].
### iii. [tex]\((s \circ g)(1.5)\)[/tex]
To find [tex]\((s \circ g)(1.5)\)[/tex], we need to compute the composition of [tex]\(s\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 1.5\)[/tex], which means evaluating [tex]\(s\)[/tex] at [tex]\(g(1.5)\)[/tex]:
1. Evaluate [tex]\(g(1.5)\)[/tex].
2. Then, evaluate [tex]\(s\)[/tex] at the result from step 1.
From the computation:
- [tex]\( g(1.5) = (1.5)^2 - 3 = 2.25 - 3 = -0.75 \)[/tex]
- [tex]\( s(-0.75) = 2(-0.75) + 1 = -1.5 + 1 = -0.5 \)[/tex]
So, the composition is:
[tex]\[ (s \circ g)(1.5) = s(g(1.5)) = s(-0.75) = -0.5 \][/tex]
Hence, [tex]\((s \circ g)(1.5) = -0.5\)[/tex].
### iv. [tex]\((g \circ s)(-4)\)[/tex]
To find [tex]\((g \circ s)(-4)\)[/tex], we need to compute the composition of [tex]\(g\)[/tex] and [tex]\(s\)[/tex] at [tex]\(x = -4\)[/tex], which means evaluating [tex]\(g\)[/tex] at [tex]\(s(-4)\)[/tex]:
1. Evaluate [tex]\(s(-4)\)[/tex].
2. Then, evaluate [tex]\(g\)[/tex] at the result from step 1.
From the computation:
- [tex]\( s(-4) = 2(-4) + 1 = -8 + 1 = -7 \)[/tex]
- [tex]\( g(-7) = (-7)^2 - 3 = 49 - 3 = 46 \)[/tex]
So, the composition is:
[tex]\[ (g \circ s)(-4) = g(s(-4)) = g(-7) = 46 \][/tex]
Hence, [tex]\((g \circ s)(-4) = 46\)[/tex].
In summary:
i. [tex]\((s \cdot g)(1) = -6\)[/tex]
ii. [tex]\((g - s)(3) = -1\)[/tex]
iii. [tex]\((s \circ g)(1.5) = -0.5\)[/tex]
iv. [tex]\((g \circ s)(-4) = 46\)[/tex]
