Help please! I got so far with this and stopped because of confusion. Here is what I have so far.
Let f(x) = 6/x-1 and g(x) = 1 + 3/x find the composite function of (fog)(x)
(fog)(x) = f (g(x))
=f (1+3/x)
=6/x-1
????
ok up to
=f (1+3/x) , then
= 6/[1+3/x - 1]
= 6/(3/x)
= 2x
So 2x would be the answer? I was thinking about including the 1+3/x-1 in the middle of the problem... So then 2x is it then I suppose?
It is a good idea to check you final answer by picking some value of x
e.g let x = 3
g(3) = 1+3/3 = 2
then f(g(3)) = f(2) = 6/(2-1) = 6
my expression was f(g(x)) = 2x
then f(g(3)) = 2(3) = 6
Even though this does not "prove" that I have the right answer, there is a very high probability that it is correct.
To find the composite function (fog)(x), we need to substitute g(x) into f(x).
Given f(x) = 6/(x-1) and g(x) = 1 + 3/x, we substitute g(x) into f(x) as follows:
(fog)(x) = f(g(x))
= f(1 + 3/x)
Now, to evaluate f(1 + 3/x), substitute 1 + 3/x into x in f(x) and simplify:
f(1 + 3/x) = 6/(1 + 3/x - 1)
= 6/(3/x)
= 6x/3
Simplifying further, we get:
= 2x
Therefore, the composite function (fog)(x) simplifies to 2x.