At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Let's break down each part of this question step by step.
### a. [tex]\( f(t) = (r + s)(t) \)[/tex]
First, we need to find [tex]\( r(t) + s(t) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Adding these functions:
[tex]\[ f(t) = r(t) + s(t) \][/tex]
[tex]\[ f(t) = (2t - 3) + (5 - 3t) \][/tex]
[tex]\[ f(t) = 2t - 3 + 5 - 3t \][/tex]
[tex]\[ f(t) = 2t - 3t + 5 - 3 \][/tex]
[tex]\[ f(t) = -t + 2 \][/tex]
So, the formula for [tex]\( f(t) \)[/tex] is:
[tex]\[ f(t) = -t + 2 \][/tex]
### b. [tex]\( g(t) = \left(\frac{s}{r}\right)(t) \)[/tex]
Now we need to find [tex]\( \frac{s(t)}{r(t)} \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Dividing these functions:
[tex]\[ g(t) = \frac{s(t)}{r(t)} \][/tex]
[tex]\[ g(t) = \frac{5 - 3t}{2t - 3} \][/tex]
So, the formula for [tex]\( g(t) \)[/tex] is:
[tex]\[ g(t) = \frac{5 - 3t}{2t - 3} \][/tex]
### c. [tex]\( h(t) = (r \cdot s)(t) \)[/tex]
Next, we multiply [tex]\( r(t) \)[/tex] and [tex]\( s(t) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Multiplying these functions:
[tex]\[ h(t) = r(t) \cdot s(t) \][/tex]
[tex]\[ h(t) = (2t - 3)(5 - 3t) \][/tex]
Using the distributive property:
[tex]\[ h(t) = 2t \cdot 5 + 2t \cdot (-3t) + (-3) \cdot 5 + (-3) \cdot (-3t) \][/tex]
[tex]\[ h(t) = 10t - 6t^2 - 15 + 9t \][/tex]
[tex]\[ h(t) = -6t^2 + 19t - 15 \][/tex]
So, the formula for [tex]\( h(t) \)[/tex] is:
[tex]\[ h(t) = -6t^2 + 19t - 15 \][/tex]
### d. [tex]\( q(t) = (s \circ r)(t) \)[/tex]
For this composition function, we need to find [tex]\( s(r(t)) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
We substitute [tex]\( r(t) \)[/tex] into [tex]\( s \)[/tex]:
[tex]\[ q(t) = s(r(t)) \][/tex]
[tex]\[ q(t) = s(2t - 3) \][/tex]
Now evaluate [tex]\( s \)[/tex] at [tex]\( 2t - 3 \)[/tex]:
[tex]\[ s(u) = 5 - 3u \][/tex]
Substitute [tex]\( u = 2t - 3 \)[/tex]:
[tex]\[ s(2t - 3) = 5 - 3(2t - 3) \][/tex]
[tex]\[ s(2t - 3) = 5 - 6t + 9 \][/tex]
[tex]\[ s(2t - 3) = 14 - 6t \][/tex]
So, the formula for [tex]\( q(t) \)[/tex] is:
[tex]\[ q(t) = 14 - 6t \][/tex]
### e. [tex]\( w(t) = r(t - 4) + 7 \)[/tex]
We need to find [tex]\( r \)[/tex] evaluated at [tex]\( t - 4 \)[/tex], then add 7.
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
Substitute [tex]\( t - 4 \)[/tex] into [tex]\( r(t) \)[/tex]:
[tex]\[ r(t - 4) = 2(t - 4) - 3 \][/tex]
[tex]\[ r(t - 4) = 2t - 8 - 3 \][/tex]
[tex]\[ r(t - 4) = 2t - 11 \][/tex]
Now, add 7 to this result:
[tex]\[ w(t) = (2t - 11) + 7 \][/tex]
[tex]\[ w(t) = 2t - 4 \][/tex]
So, the formula for [tex]\( w(t) \)[/tex] is:
[tex]\[ w(t) = 2t - 4 \][/tex]
Thus, the simplified formulas are:
a. [tex]\( f(t) = -t + 2 \)[/tex]
b. [tex]\( g(t) = \frac{5 - 3t}{2t - 3} \)[/tex]
c. [tex]\( h(t) = -6t^2 + 19t - 15 \)[/tex]
d. [tex]\( q(t) = 14 - 6t \)[/tex]
e. [tex]\( w(t) = 2t - 4 \)[/tex]
Given these formulas, if we compute at some t-values we get:
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( g(1) = -2.0 \)[/tex]
- [tex]\( h(2) = -1 \)[/tex]
- [tex]\( q(3) = -4 \)[/tex]
- [tex]\( w(4) = 4 \)[/tex]
### a. [tex]\( f(t) = (r + s)(t) \)[/tex]
First, we need to find [tex]\( r(t) + s(t) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Adding these functions:
[tex]\[ f(t) = r(t) + s(t) \][/tex]
[tex]\[ f(t) = (2t - 3) + (5 - 3t) \][/tex]
[tex]\[ f(t) = 2t - 3 + 5 - 3t \][/tex]
[tex]\[ f(t) = 2t - 3t + 5 - 3 \][/tex]
[tex]\[ f(t) = -t + 2 \][/tex]
So, the formula for [tex]\( f(t) \)[/tex] is:
[tex]\[ f(t) = -t + 2 \][/tex]
### b. [tex]\( g(t) = \left(\frac{s}{r}\right)(t) \)[/tex]
Now we need to find [tex]\( \frac{s(t)}{r(t)} \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Dividing these functions:
[tex]\[ g(t) = \frac{s(t)}{r(t)} \][/tex]
[tex]\[ g(t) = \frac{5 - 3t}{2t - 3} \][/tex]
So, the formula for [tex]\( g(t) \)[/tex] is:
[tex]\[ g(t) = \frac{5 - 3t}{2t - 3} \][/tex]
### c. [tex]\( h(t) = (r \cdot s)(t) \)[/tex]
Next, we multiply [tex]\( r(t) \)[/tex] and [tex]\( s(t) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
Multiplying these functions:
[tex]\[ h(t) = r(t) \cdot s(t) \][/tex]
[tex]\[ h(t) = (2t - 3)(5 - 3t) \][/tex]
Using the distributive property:
[tex]\[ h(t) = 2t \cdot 5 + 2t \cdot (-3t) + (-3) \cdot 5 + (-3) \cdot (-3t) \][/tex]
[tex]\[ h(t) = 10t - 6t^2 - 15 + 9t \][/tex]
[tex]\[ h(t) = -6t^2 + 19t - 15 \][/tex]
So, the formula for [tex]\( h(t) \)[/tex] is:
[tex]\[ h(t) = -6t^2 + 19t - 15 \][/tex]
### d. [tex]\( q(t) = (s \circ r)(t) \)[/tex]
For this composition function, we need to find [tex]\( s(r(t)) \)[/tex].
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
[tex]\[ s(t) = 5 - 3t \][/tex]
We substitute [tex]\( r(t) \)[/tex] into [tex]\( s \)[/tex]:
[tex]\[ q(t) = s(r(t)) \][/tex]
[tex]\[ q(t) = s(2t - 3) \][/tex]
Now evaluate [tex]\( s \)[/tex] at [tex]\( 2t - 3 \)[/tex]:
[tex]\[ s(u) = 5 - 3u \][/tex]
Substitute [tex]\( u = 2t - 3 \)[/tex]:
[tex]\[ s(2t - 3) = 5 - 3(2t - 3) \][/tex]
[tex]\[ s(2t - 3) = 5 - 6t + 9 \][/tex]
[tex]\[ s(2t - 3) = 14 - 6t \][/tex]
So, the formula for [tex]\( q(t) \)[/tex] is:
[tex]\[ q(t) = 14 - 6t \][/tex]
### e. [tex]\( w(t) = r(t - 4) + 7 \)[/tex]
We need to find [tex]\( r \)[/tex] evaluated at [tex]\( t - 4 \)[/tex], then add 7.
Given:
[tex]\[ r(t) = 2t - 3 \][/tex]
Substitute [tex]\( t - 4 \)[/tex] into [tex]\( r(t) \)[/tex]:
[tex]\[ r(t - 4) = 2(t - 4) - 3 \][/tex]
[tex]\[ r(t - 4) = 2t - 8 - 3 \][/tex]
[tex]\[ r(t - 4) = 2t - 11 \][/tex]
Now, add 7 to this result:
[tex]\[ w(t) = (2t - 11) + 7 \][/tex]
[tex]\[ w(t) = 2t - 4 \][/tex]
So, the formula for [tex]\( w(t) \)[/tex] is:
[tex]\[ w(t) = 2t - 4 \][/tex]
Thus, the simplified formulas are:
a. [tex]\( f(t) = -t + 2 \)[/tex]
b. [tex]\( g(t) = \frac{5 - 3t}{2t - 3} \)[/tex]
c. [tex]\( h(t) = -6t^2 + 19t - 15 \)[/tex]
d. [tex]\( q(t) = 14 - 6t \)[/tex]
e. [tex]\( w(t) = 2t - 4 \)[/tex]
Given these formulas, if we compute at some t-values we get:
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( g(1) = -2.0 \)[/tex]
- [tex]\( h(2) = -1 \)[/tex]
- [tex]\( q(3) = -4 \)[/tex]
- [tex]\( w(4) = 4 \)[/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
