gaussian_solution.tex

\begin{itemized}

\item $L(x) = f(x) g(h(q(r(x))))$, where $f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \frac{1}{x}$, 
  $g(x) = e^x$, $h(x) = -\frac12 x^2$, $q(x) = \frac{x - \mu}{\sigma}$, and $r(x) = \log(x)$.
  
\item \texttt{f = function(x) exp(-.5*log(x)^2)/x/sqrt(2*pi)}

\item \texttt{curve(f, 0, 5)}

\item No. We can see this on the graph. Analytically, both $L(.136)$ and $L(.999)$ are roughly equal to .4.

\item $f(g(x)) = f(g(y)) \Longleftrightarrow g(x) = g(y) \Longleftrightarrow x = y$. $f, g, q$, and $r$ are 1-1, but $h$ is not. 

\end{itemized}