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) X g(x), g(x) X f(x), and f(x) X f(x), we need to understand what these operations mean and how to perform them.
The notation "f X g" represents the composition of functions f and g. This means that we evaluate the output of g and use it as the input for f.
Let's calculate each expression following these steps.
1. f(x) X g(x):
To find f(g(x)), we substitute g(x) into the function f(x):
f(g(x)) = 3√(g(x)-1)
Substitute g(x) = x^3 + 1:
f(g(x)) = 3√((x^3 + 1) - 1)
= 3√(x^3)
So, f(g(x)) simplifies to 3x√(x).
2. g(x) X f(x):
To find g(f(x)), we substitute f(x) into the function g(x):
g(f(x)) = (f(x))^3 + 1
Substitute f(x) = 3√(x-1):
g(f(x)) = ((3√(x-1))^3) + 1
= (3√(x-1))^3 + 1
= 27(x-1) + 1
= 27x - 26
Therefore, g(f(x)) simplifies to 27x - 26.
3. f(x) X f(x):
To find f(f(x)), we substitute f(x) into the function f(x):
f(f(x)) = 3√(f(x)-1)
Substitute f(x) = 3√(x-1):
f(f(x)) = 3√(3√(x-1)-1)
This expression cannot be simplified further.
So, the correct answers are:
f(g(x)) = 3x√(x)
g(f(x)) = 27x - 26
f(f(x)) = 3√(3√(x-1)-1)
Make sure to verify these results and check with your teacher for confirmation.