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
f(x)= cubrt(x-1)
g(x)=x^3+1
f(g(x))= cubrt(x^3+1-1)=x
g(f(x)= f(x)^3+1=(cubrt(x-1)^3-1=x
f(f)= cubr(cubrt(x-1)-1) which might be simplified, but I don't see it now.
To find the compositions f(x)∘g(x) and g(x)∘f(x), which are denoted as (a) f X g and (b) g X f respectively, you need to substitute the function g(x) into f(x) and vice versa.
Let's start by finding f X g:
To calculate f(g(x)), substitute g(x) into f(x), which means you replace every x in f(x) with g(x):
f(g(x)) = 3√(g(x) - 1)
Since g(x) = x^3 + 1, we can substitute it into the equation above:
f(g(x)) = 3√((x^3 + 1) - 1)
Simplifying further:
f(g(x)) = 3√(x^3)
f(g(x)) = 3x^(3/2)
Therefore, (a) f X g = 3x^(3/2).
Now let's find g X f:
To calculate g(f(x)), we substitute f(x) into g(x):
g(f(x)) = (f(x))^3 + 1
Since f(x) = 3√(x - 1), we can substitute it into the equation above:
g(f(x)) = ((3√(x - 1))^3) + 1
Remember that raising a cube root to the power of 3 simply cancels out the cube root:
g(f(x)) = (x - 1) + 1
Simplifying further:
g(f(x)) = x
Therefore, (b) g X f = x.
Lastly, to find f X f, you need to substitute f(x) into itself:
f(f(x)) = 3√(f(x) - 1)
Since f(x) = 3√(x - 1), substitute it into the equation above:
f(f(x)) = 3√(3√(x - 1) - 1)
Simplifying further can be a bit complicated algebraically, but there is no simplification rule that would lead to f X f = 6√(x + 1).
Hence, the correct answer for (c) f X f is the expression itself: f(f(x)) = 3√(f(x) - 1).