Final answer:
It is false that Fanny Mendelssohn Hensel was encouraged by her family to pursue a musical career; in contrast, her family discouraged her from seeking professional recognition.
Step-by-step explanation:
The statement that Fanny Mendelssohn Hensel was enthusiastically encouraged to pursue a musical career by her family is False. Fanny Mendelssohn Hensel, the sister of the well-known composer Felix Mendelssohn, faced significant obstacles in her musical career due to the gender norms of her time. Although she was a prodigiously talented musician and composer, her family, especially her father and brother, believed that a career in music was not proper for a woman of her standing. They discouraged her from publishing and gaining professional recognition for her work. Despite this, Mendelssohn Hensel composed over 460 pieces of music, although most of her work was published posthumously or under her brother's name.