Explanation:
Let's find the compositions of the functions f(t) and g(t):
a) f(g(t)):
f(g(t)) means you first find g(t) and then use the result as the input for f(t).
g(t) = 5t + 1
Now, substitute g(t) into f(t):
f(g(t)) = f(5t + 1)
f(t) = 9t²
Replace t with (5t + 1) in f(t):
f(g(t)) = 9(5t + 1)²
Now, simplify:
f(g(t)) = 9(25t² + 10t + 1)
f(g(t)) = 225t² + 90t + 9
b) g(f(t)):
g(f(t)) means you first find f(t) and then use the result as the input for g(t).
f(t) = 9t²
Now, substitute f(t) into g(t):
g(f(t)) = 5(9t²) + 1
g(f(t)) = 45t² + 1
c) f(f(t)):
f(f(t)) means you use f(t) as the input for f(t) itself.
f(t) = 9t²
Now, replace t with 9t² in f(t):
f(f(t)) = 9(9t²)²
Simplify:
f(f(t)) = 9(81t⁴)
f(f(t)) = 729t⁴
d) g(g(t)):
g(g(t)) means you use g(t) as the input for g(t) itself.
g(t) = 5t + 1
Now, replace t with (5t + 1) in g(t):
g(g(t)) = 5(5t + 1) + 1
Simplify:
g(g(t)) = 25t + 5 + 1
g(g(t)) = 25t + 6