The phrase "assumed the task of" is correct and usable in written English.
You can use it when describing someone taking on a responsibility or duty in a particular context.
Example: "After the previous manager left, Sarah assumed the task of leading the project team."
Alternatives: "took on the responsibility of" or "accepted the role of".
Reeve assumed the task of caring for her mother until the older woman's death, in February 2001.
As a young instructor, he assumed the task of writing a series of laboratory manuals for the Duke biology department.
Shuhei Yoshida, president of Sony Worldwide Studios, assumed the task of unveiling the big guns which, Sony hopes, will persuade punters to buy PlayStation 4s this Christmas.
Three years ago, a syndicate of Asian investors, led by Henry Cheng, of Hong Kong's New World Development Company, assumed the task of arranging construction financing.
He also assumed the task of repressing the French underground movement, whose activities hampered the delivery of supplies and men to Germany.
Amid the mass movement to bring democracy to a city under Chinese rule, the eighteen-year-old student at the Hong Kong Design Institute had assumed the task of guarding protesters against rain, tear gas, and heatstroke.
Ibn Taimaya argued that that religious scholars could assume the task of declaring holy war if their rulers failed to.
The writer who assumes the task of making a fictional character of someone whose life took place in history faces particular challenges.
The theologians had to assume the task of explaining the relation between God, the unaffected and unchanging cause of all things, and the universe.
That turned out to be tricky, but the team identified an element of the product's immediate environment that could assume the task of the instruction sheet: The instructions were printed on the product packaging.
Besides assuming the task of having to defend Michael Jordan in the first game of his comeback, Anderson had an 0-for-5 shooting night that might have even been worse than the numbers indicate.
