Sure, let's solve the given problem step by step.
Step 1: Identify the functions
We're given two functions, f(x) = (1-x)/(1+x) and g(x) = x/(1+x).
Step 2: Composition of functions (f[g(x)])
We need to find the composite function f(g(x)). To form the composite function f[g(x)], we take the function g(x) and substitute it into function f(x) for every instance of 'x'.
Here, g(x) = x/(1+x). Substituting g(x) into f(x), we get f(g(x)) = (1-g(x))/(1+g(x)).
Step 3: Substitution
Now, we replace g(x) with its exact expression which is x/(1+x). So, the function becomes f(g(x)) = (1-(x/(1+x)))/(1+(x/(1+x))).
Step 4: Simplification
Next, let's simplify the above expression. On simplifying, we obtain f(g(x)) = 1/(2x + 1). So, the simplified form of f[g(x)] is 1/(2x + 1).
Step 5: Domain
The domain of a function includes all the possible values of x for which the function is defined. The function g(x) = x/(1+x) will be undefined when if the denominator becomes zero. The denominator of g(x) becomes zero when x=-1. Therefore, the domain of the compiled function f[g(x)] is the same as the domain of g(x), which is all real numbers except x = -1.
Therefore, the simplified form for f[g(x)] for the functions f(x) = (1-x)/(1+x) and g(x) = x/(1+x) is 1/(2x + 1) and its domain is all real numbers except -1.