Answer
$ f(f(a))=\frac{a+1}{a+2}$.
Work Step by Step
The given function is $f(x)=\frac{1}{x+1}$.
Replace $x$ with $a$.
$\Rightarrow f(a)=\frac{1}{a+1}$
Replace $a$ with $f(a)$.
$\Rightarrow f(f(a))=\frac{1}{f(a)+1}$
Substitute the value of $f(a)$ on the right hand side.
$\Rightarrow f(f(a))=\frac{1}{\frac{1}{a+1}+1}$
Solve the denominator.
$\Rightarrow f(f(a))=\frac{1}{\frac{1}{a+1}+\frac{1}{1}}$
The LCD of the denominator is $(a+1)$.
$\Rightarrow f(f(a))=\frac{1}{\frac{1}{a+1}+\frac{a+1}{a+1}}$
Add numerators because denominators are equal.
$\Rightarrow f(f(a))=\frac{1}{\frac{1+a+1}{a+1}}$
Simplify.
$\Rightarrow f(f(a))=\frac{1}{\frac{a+2}{a+1}}$
Invert the division and multiply.
$\Rightarrow f(f(a))=1\cdot \frac{a+1}{a+2}$
Cancel common terms.
$\Rightarrow f(f(a))=\frac{a+1}{a+2}$.
