I'm wondering if I could use the Acceptance-Rejection Algorithm to sample from the Exponential distribution, or from some random variable $X$ dist on $(\lambda)$, such that $f(x; \lambda)=\lambda e^{-\lambda x}$.
Choosing the "easy" function to sample from, $g(x): g(x) = e^x$, I'm now faced finding $c \in \mathbb{R}$ such that $$h(x) = \frac{f(x)}{cg(x)}$$ has been maximized.
However, $h(x) = c^{-1} \lambda e^{x(1-\lambda)}$ does not have a maximum; solving for $h'(x) = 0$ doesn't yield any solution.
Textbooks and notes I've seen suggest that we want $c$ to be as close to $1$ as possible. In this case, why couldn't I just say $c=1$?
If I can't find a $c$, should I be using another method altogether, i.e. inverse function?