To find the simplified form for f[g(x)], we need to substitute g(x) into the expression for f(x) and simplify.
First, let's find g(x) by substituting x into the expression for g(x):
g(x) = x/(1 + x)
Next, substitute g(x) into the expression for f(x):
f[g(x)] = f[x/(1 + x)]
Now, let's substitute the expression for g(x) into f(x):
f[g(x)] = f[x/(1 + x)] = (1 - (x/(1 + x)))/(1 + (x/(1 + x)))
To simplify, we can simplify the numerator and denominator separately:
Numerator:
1 - (x/(1 + x)) = (1 + x - x)/(1 + x) = 1/(1 + x)
Denominator:
1 + (x/(1 + x)) = (1 + x + x)/(1 + x) = (1 + 2x)/(1 + x)
Putting it all together, the simplified form of f[g(x)] is:
f[g(x)] = 1/(1 + x)/(1 + 2x)/(1 + x)
Now, let's simplify further by multiplying the numerator and denominator by the reciprocal of the denominator:
f[g(x)] = (1/(1 + x)) * ((1 + x)/(1 + 2x))
Canceling out the common terms in the numerator and denominator, we get:
f[g(x)] = 1/(1 + 2x)
The simplified form of f[g(x)] is 1/(1 + 2x).
Now, let's consider the domain. The only restriction in the domain arises when the denominator becomes zero since division by zero is undefined. Therefore, we need to find the values of x that make the denominator equal to zero:
1 + 2x = 0
Solving for x gives:
2x = -1
x = -1/2
So, the domain of f[g(x)] is all real numbers except x = -1/2.