Question: 2018 Winter Midterm - 12
The functions $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow \mathbb{N}$ are
recursively defined as follows:
$$
\begin{alignat}{2}
f(0) &= 0, \\
f(n) &= 2 + f(n - 1)\ \; &\mathrm{if}\ n \geq 1, \\
g(0) &= 1, \\
g(n) &= 7 \cdot g(n - 1)\ \; &\mathrm{if}\ n \geq 1.
\end{alignat}
$$
For any integer $n \geq 0$, what is $g(f(n))$?
(a)
$7^{n}$
(b)
$n^{7}$
(c)
$7^{2n}$
(d)
$(2n)^{7}$