Rahim was the youngest tennis player ever to play for Pakistan in the Davis Cup at 15 years of age.

His father, Mir Abdur Rahim, was a civil servant who encouraged his children, sons and daughters, to play tennis.

In 1970 he won the U.S. National Amateur Grass Court Championships and the March of Dimes Open against Jimmy Connors.

Rahim also won two ATP titles, the first at the Arkansas International in Little Rock against former Wimbledon runner up Alex Metreveli of the Soviet Union and then the second at National Tennis Foundation Open in Cleveland against Colin Dibley, both in 1976.

[2] Rahim is the winner of one of the closest matches ever played when he beat Tom Gorman 6–7(3–5), 7–6(5–1), 7–6(5–4) at the Pennsylvania Lawn Tennis Championships.