Find the domain of the composite function f of g.
f(x)=3/x-6; g(x)= 1/x
please help and show the work as well.
f.g=1/(3)*(x-6)=x/3 - 2
domain all x
To find the domain of the composite function f(g(x)), we need to consider two things:
1. The domain of the function g(x)
2. The domain of the function f(x) after substituting g(x) into it.
First, let's determine the domain of g(x) = 1/x. The domain of a rational function like this is all real numbers except for the value(s) of x that make the denominator zero. In this case, the denominator (x) should not be equal to zero.
So, the domain of g(x) is all real numbers except x = 0.
Next, we substitute g(x) = 1/x into f(x) = 3/(x - 6). Replace every instance of x in f(x) with g(x):
f(g(x)) = 3/(g(x) - 6)
Replace g(x) with 1/x:
f(g(x)) = 3/(1/x - 6)
Now we simplify the expression:
f(g(x)) = 3/((1 - 6x)/x)
To simplify further, we multiply the fraction by the reciprocal of the denominator:
f(g(x)) = 3 * (x/(1 - 6x))
The domain of f(g(x)) will be any real number for which the expression (1 - 6x) is not zero and the denominator (1 - 6x) is not zero.
So, (1 - 6x) ≠ 0
Solving for x, we have:
1 - 6x ≠ 0
-6x ≠ -1
x ≠ 1/6
Therefore, the domain of the composite function f(g(x)) is all real numbers except x = 0 and x = 1/6.