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]
