 Solution

Integrating the two linear pieces gives the cumulative distribution functionThe split probability is $F(3)=1/3$. Invert each branch to obtain the method of inversion:Both expressions equal three at the split. For an ideal uniform draw, monotonicity gives $\mathbb{P}(F^{-1}(U)\le x)=\mathbb{P}(U\le F(x))=F(x)$, proving the target law. Inversion is preferable here: its quantile is explicit, it uses one uniform and one square root per output, and has no rejected draws. Rejection remains useful for densities with no convenient inverse CDF.