From simple questions to complex issues, IDNLearn.com has the answers you need. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
