Find f X g , g X f , and f X f

f(x) = 3sqrt x-1 g(x) = x^3 + 1

For f(x) the 3 is little above the sqrt sign.

My answers were:

f(g(x)) = 3sqrt x^3 + 1 -1
g(f(x)) = 3sqrt (x-1)^3 -1
f(f) = 3sqrt x-1-1

My teacher said these were wrong. Please help. I do not know how to correct these. Thank you!

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.