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, g(X)f, and f(X)f, you need to apply the compositions of functions. Let's go through each case step by step:
1. f(X)g:
To find f(X)g, you need to substitute g(x) into f(x). In other words, replace every occurrence of "x" in f(x) with the expression g(x).
Given:
f(x) = 3√(x-1)
g(x) = x^3 + 1
Substituting g(x) into f(x):
f(g(x)) = 3√((x^3 + 1) - 1)
Simplifying further:
f(g(x)) = 3√(x^3)
f(g(x)) = 3x^(3/2)
So, f(X)g is equal to 3x^(3/2).
2. g(X)f:
To find g(X)f, you need to substitute f(x) into g(x). In other words, replace every occurrence of "x" in g(x) with the expression f(x).
Given:
f(x) = 3√(x-1)
g(x) = x^3 + 1
Substituting f(x) into g(x):
g(f(x)) = (3√(x-1))^3 + 1
Simplifying further:
g(f(x)) = (x-1)^3 + 1
So, g(X)f is equal to (x-1)^3 + 1.
3. f(X)f:
To find f(X)f, you need to substitute f(x) into f(x). In other words, replace every occurrence of "x" in f(x) with the expression f(x).
Given:
f(x) = 3√(x-1)
Substituting f(x) into f(x):
f(f(x)) = 3√(3√(x-1)-1)
So, f(X)f is equal to 3√(3√(x-1)-1).
Please check these solutions against your teacher's expectations.