Let X ( t ) be a continuously time-changed Brownian motion starting from a random position η , S ( t ) a given continuous, increasing boundary, with S ( 0 ) ≥ 0 , P ( η ≥ S ( 0 ) ) = 1 , and F an assigned distribution function. We study the inverse first-passage time problem for X ( t ) , which consists in finding the distribution of η such that the first-passage time of X ( t ) below S ( t ) has distribution F , generalizing the results, valid in the case when S ( t ) is a straight line. Some explicit examples are reported.