Calculate the composite functions f∘g and g∘f.
f(x)=6x,g(x)=x8
f(g(x))=
g(f(x))=
f(g) = 6g = 6x^8
g(f) = f^8 = (6x)^8
To calculate the composite functions f∘g and g∘f, we need to substitute the function inside the parentheses into the other function and simplify the resulting expression. Let's start with f(g(x)).
1. f(g(x)) means we substitute g(x) into the function f(x):
f(g(x)) = f(x^8)
Now let's simplify the expression further. Given that f(x) = 6x:
2. Replace x in f(x^8) with x^8:
f(x^8) = 6(x^8)
So, f(g(x)) simplifies to:
f(g(x)) = 6(x^8)
Next, let's calculate g∘f.
1. g(f(x)) means we substitute f(x) into the function g(x):
g(f(x)) = g(6x)
Given that g(x) = x^8:
2. Replace x in g(6x) with 6x:
g(6x) = (6x)^8
Simplifying further:
g(f(x)) = (6x)^8
Therefore, the composite functions are:
f(g(x)) = 6(x^8)
g(f(x)) = (6x)^8