Answer
$\displaystyle \qquad f(x)=\frac{6}{1+2^{-x}} $
Work Step by Step
Goal: $\displaystyle \quad f(x)=\frac{N}{1+Ab^{-x}}$,
where $N$ = limiting value = $6$ (given)
$f(0)=3\Rightarrow\left\{\begin{array}{ll}
3 & =\dfrac{6}{1+Ab^{0}}\\
1+A & =6/3\\
A & =2-1=1
\end{array}\right.\qquad\Rightarrow\qquad A=1$
$f(x)=\displaystyle \frac{6}{1+b^{-x}}$
$f(1)=4\Rightarrow\left\{\begin{array}{ll}
4 & =\frac{6}{1+b^{-1}}\\
& \\
1+b & =6/4\\
b^{-1} & =3/2-1=1/2\\
b & =2
\end{array}\right.$
Thus, $\displaystyle \qquad f(x)=\frac{6}{1+2^{-x}} $
