Question22. If $f(x)=\frac{1}{1+x}$ and $g(x)=\frac{1}{2+x}$, determine $f(g(x))$.

Studdy Solution
Simplify the expression $\frac{1}{1 + \frac{1}{2+x}}$. To do this, we need to simplify the denominator:
1. Start with the expression inside the denominator: $1 + \frac{1}{2+x}$.
2. Find a common denominator for the terms in the denominator:
$$1 + \frac{1}{2+x} = \frac{(2+x)}{(2+x)} + \frac{1}{2+x} = \frac{(2+x) + 1}{2+x} = \frac{3+x}{2+x}$$
3. Substitute back into the main expression:
$$f(g(x)) = \frac{1}{\frac{3+x}{2+x}}$$
4. Simplify by taking the reciprocal of the fraction in the denominator:
$$f(g(x)) = \frac{2+x}{3+x}$$
The expression for $f(g(x))$ is $\boxed{\frac{2+x}{3+x}}$.