Question: 2018 Fall Midterm - 12
The functions $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow \mathbb{N}$ are
recursively defined as follows:
$$
\begin{alignat}{2}
f(0) &= 3, \\
f(n) &= 5 + f(n - 1)\ \; &\mathrm{if}\ n \geq 1, \\
g(0) &= 1, \\
g(n) &= 2 \cdot g(n - 1) &\mathrm{if}\ n \geq 1.
\end{alignat}
$$
For any integer $n \geq 0$, what is $f(g(n))$?
(a)
$2^{3 + 5n}$
(b)
$5 + 3 \cdot 2^{n}$
(c)
$2^{5 + 3n}$
(d)
$3 + 5 \cdot 2^{n}$