To find f X g, you need to substitute g(x) into f(x). Let's start by replacing x in f(x) with g(x):
f(g(x)) = 3√(g(x) - 1)
Since g(x) = x^3 + 1, we can substitute it:
f(g(x)) = 3√((x^3 + 1) - 1)
Next, simplify the expression inside the square root:
f(g(x)) = 3√(x^3)
Now, let's move on to g X f:
g(f(x)) = (f(x))^3 + 1
Since f(x) = 3√(x - 1), we substitute it into g(x):
g(f(x)) = (3√(x - 1))^3 + 1
To simplify further, we cube the expression inside the parentheses:
g(f(x)) = 27(x - 1) + 1
Expand:
g(f(x)) = 27x - 27 + 1
g(f(x)) = 27x - 26
Finally, to find f X f, we substitute f(x) into f(x):
f(f(x)) = 3√(f(x) - 1)
Since f(x) = 3√(x - 1), we replace f(x) with this expression:
f(f(x)) = 3√(3√(x - 1) - 1)
These are the correct forms of f X g, g X f, and f X f.