Solution
= Solution
{c}
Solving $u=F(x)$ gives $1-(1-p)e^{-\beta x}=p^{1-u}$. Hence the <quantile function> is
$$
\boxed{F^{-1}(u)=-\frac1\beta\log\!\left(\frac{1-p^{1-u}}{1-p}\right),\qquad0<u<1.}
$$
For such $u$, $p<p^{1-u}<1$, making the <logarithm>'s argument lie in $(0,1)$ and the output positive. Its endpoint limits are $F^{-1}(0)=0$ and $F^{-1}(1)=+\infty$. Substitution into (i)'s <cumulative distribution function> recovers $u$.