Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
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]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
