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Using [tex]$f(t)=3t^2$[/tex] and [tex]$g(t)=8t+1$[/tex], find:

(a) [tex]f(g(t))=[/tex] [tex]$\square$[/tex]

(b) [tex]g(f(t))=[/tex] [tex]$\square$[/tex]

(c) [tex]f(f(t))=[/tex] [tex]$\square$[/tex]

(d) [tex]g(g(t))=[/tex] [tex]$\square$[/tex]

Sagot :

GinnyAnswer
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]
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