Let's solve each part of the problem step-by-step.
Given the functions [tex]\( f(t) = 3t^2 \)[/tex] and [tex]\( g(t) = 8t + 1 \)[/tex].
(a) Find [tex]\( f(g(t)) \)[/tex]:
1. Start by substituting [tex]\( g(t) \)[/tex] into [tex]\( f(t) \)[/tex].
2. So, [tex]\( f(g(t)) = f(8t + 1) \)[/tex].
3. To find this, replace [tex]\( t \)[/tex] in [tex]\( f(t) = 3t^2 \)[/tex] with [tex]\( 8t + 1 \)[/tex]:
[tex]\[ f(8t + 1) = 3(8t + 1)^2 \][/tex].
Thus, [tex]\( f(g(t)) = 3(8t + 1)^2 \)[/tex].
(b) Find [tex]\( g(f(t)) \)[/tex]:
1. Start by substituting [tex]\( f(t) \)[/tex] into [tex]\( g(t) \)[/tex].
2. So, [tex]\( g(f(t)) = g(3t^2) \)[/tex].
3. To find this, replace [tex]\( t \)[/tex] in [tex]\( g(t) = 8t + 1 \)[/tex] with [tex]\( 3t^2 \)[/tex]:
[tex]\[ g(3t^2) = 8(3t^2) + 1 \][/tex].
[tex]\[ g(3t^2) = 24t^2 + 1 \][/tex].
Thus, [tex]\( g(f(t)) = 24t^2 + 1 \)[/tex].
(c) Find [tex]\( f(f(t)) \)[/tex]:
1. Start by substituting [tex]\( f(t) \)[/tex] into itself.
2. So, [tex]\( f(f(t)) = f(3t^2) \)[/tex].
3. To find this, replace [tex]\( t \)[/tex] in [tex]\( f(t) = 3t^2 \)[/tex] with [tex]\( 3t^2 \)[/tex]:
[tex]\[ f(3t^2) = 3(3t^2)^2 \][/tex].
[tex]\[ f(3t^2) = 3(9t^4) \][/tex].
[tex]\[ f(3t^2) = 27t^4 \][/tex].
Thus, [tex]\( f(f(t)) = 27t^4 \)[/tex].
(d) Find [tex]\( g(g(t)) \)[/tex]:
1. Start by substituting [tex]\( g(t) \)[/tex] into itself.
2. So, [tex]\( g(g(t)) = g(8t + 1) \)[/tex].
3. To find this, replace [tex]\( t \)[/tex] in [tex]\( g(t) = 8t + 1 \)[/tex] with [tex]\( 8t + 1 \)[/tex]:
[tex]\[ g(8t + 1) = 8(8t + 1) + 1 \][/tex].
[tex]\[ g(8t + 1) = 64t + 8 + 1 \][/tex].
[tex]\[ g(8t + 1) = 64t + 9 \][/tex].
Thus, [tex]\( g(g(t)) = 64t + 9 \)[/tex].
Putting it all together, we have:
(a) [tex]\( f(g(t)) = 3(8t + 1)^2 \)[/tex]
(b) [tex]\( g(f(t)) = 24t^2 + 1 \)[/tex]
(c) [tex]\( f(f(t)) = 27t^4 \)[/tex]
(d) [tex]\( g(g(t)) = 64t + 9 \)[/tex]
