Find (a) f X g , (b) g X f, and (c) f X f
f(x) = 3sqrt x-1 g(x) = x^3 + 1
For the f(x) the 3 is little above the sqrt sign.
my answers were:
(a) = 3sqrt x^4 -1
(b) = x^4 -1
(c) = 6sqrt x+1
These answers are wrong but I do not know how to do this.
To find the composition of two functions, denoted as f(x) ∘ g(x) or f X g, you need to substitute the second function (g(x)) into the first function (f(x)) and simplify. Similarly, to find g X f, you'll substitute f(x) into g(x) and simplify. Lastly, to find f X f, you'll substitute f(x) into itself.
Let's calculate each of these compositions step by step:
(a) f X g:
To find f X g, substitute g(x) into f(x):
f(g(x)) = f(x^3 + 1)
Now, replace f(x) with its expression:
f(x^3 + 1) = 3√(x^3 + 1 - 1)
Simplify the expression inside the square root:
3√(x^3) = 3x√x
Therefore, the composition f X g is equal to 3x√x.
(b) g X f:
To find g X f, substitute f(x) into g(x):
g(f(x)) = g(3√(x-1))
Now, replace g(x) with its expression:
g(3√(x-1)) = (3√(x-1))^3 + 1
Expand the cube of the square root:
(3√(x-1))^3 = 3^3 * √((x-1)^3) = 27√(x-1)^3
Finally, add 1 to the expression:
(27√(x-1)^3) + 1 = 27√(x-1)^3 + 1
Therefore, the composition g X f is equal to 27√(x-1)^3 + 1.
(c) f X f:
To find f X f, substitute f(x) into f(x):
f(f(x)) = f(3√(x-1))
Now, replace f(x) with its expression:
f(3√(x-1)) = 3√(3√(x-1)-1)
Simplify the expression inside the square root:
3√(3√(x-1)-1)
Therefore, the composition f X f is equal to 3√(3√(x-1)-1).
Your answers were incorrect because the expressions were not substituted correctly. Pay close attention to substituting the entire function (including x) instead of neglecting the variable.